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*Proof (continued):*

- We prove (
`x`subset`y`/\`y`subset`x`) =>`x`=`y`. Assume`x`subset`y`/\`y`subset`x`, i.e., by definition of ` subset '

We prove(1) **forall**`z`in`x`:`z`in`y`;(2) **forall**`z`in`y`:`z`in`x`.`x`=`y`, i.e., by definition of `=',**forall**`z`:`z`in`x`<=>`z`in`y`. Take arbitrary`z`. We have to prove`z`in`x`<=>`z`in`y`.- We prove
`z`in`x`=>`z`in`y`. Assume (3)`z`in`x`. We have to prove`z`in`y`which is a consequence of (1) and (3). - We prove
`z`in`y`=>`z`in`x`. Assume (4)`z`in`y`. We have to prove`z`in`x`which is a consequence of (2) and (4).

- We prove

*A well-structured argument based on definitions and given knowledge.*

Author: Wolfgang Schreiner

Last Modification: October 14, 1999