Definition - A label is a name.

      Definition - The term next is synonymous with the phrase immediately subsequent.

      Definition - P is a procedure if and only if each following statement is true:

      P is a collection of labeled statements.

      Exactly a single statement in P is labeled Start.

      Optionally, the Start statement may name a source or sources of input to procedure P.

      Every statement in P specifies a single action or more than a single action to be done.

      The statement in P that is labeled Start specifies the actions that are to be done initially.

      Procedure P names the labeled statement to be done next with the sentence or phrase "X is next" or "Otherwise, Y is next", where X or Y is the label of the statement to be done next; or instead, the current statement may end P by ending with the word END.

      Any statement in P may start or resume another previously defined procedure, Q, by stating the sentence or phrase "X in procedure Q is next." where X is a label in Q.

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      In case it is not clear, naming a thing is an action.

      Definition - Set D is a copy of set E if and only if D differs from E in its name, and only in its name.

      Definition - A pair of sets, p and q, have a one-to-one correspondence with each other if and only if when p and q are the inputs to the 1 to 1 procedure, this procedure is ended by its statement labeled RY.

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1 to 1 Procedure - The procedure named 1 to 1 is a procedure that tests whether a pair of sets, p and q, have a one-to-one correspondence with each other.

Input - Sets p and q are the inputs of the 1 to 1 procedure.

Start - Sets p and q are the inputs to the this procedure.

AA - If p is empty and q is empty, RY is next. Otherwise, CT is next.

CT - If p is empty, and q is not empty, RX is next. Otherwise, CD is next.

CD - If p is not empty and q is empty, RX is next. Otherwise, BX is next.

BX - Remove a single element from p, and remove a single element from q. AA is next.

RX - End.

RY - End.

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Proof 1

      Given that p and q have a one-to-one correspondence with each other, either both p and q are both empty; or for each element removed from p by the 1 to 1 procedure, exactly a single element is removed from q by this procedure.

      Given that q and r have a one-to-one correspondence with each other, either both q and r are both empty, or for each element removed from q by the 1 to 1 procedure, exactly a single element is removed from r by this procedure.

      In such a case, either p, q, and r are each empty, or for each element removed from p by the 1 to 1 procedure, exactly a single element is removed from q, and exactly a single element is removed from r by this procedure.

      Therefore, if sets p and q have a one-to-one correspondence with each other, and sets q and r have a one-to-one correspondence with each other, then set p and r have a one-to-one correspondence with each other.

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