let x, y be set ; ( <*x,y*> is constant implies x = y )
A1: rng <*x,y*> =
rng (<*x*> ^ <*y*>)
by FINSEQ_1:def 9
.=
(rng <*x*>) \/ (rng <*y*>)
by FINSEQ_1:31
.=
(rng <*x*>) \/ {y}
by FINSEQ_1:38
.=
{x} \/ {y}
by FINSEQ_1:38
.=
{x,y}
by ENUMSET1:1
;
A2:
y in {x,y}
by TARSKI:def 2;
assume
<*x,y*> is constant
; x = y
then reconsider s = <*x,y*> as constant Function ;
A3:
x in {x,y}
by TARSKI:def 2;
rng s is trivial
;
hence
x = y
by A1, A3, A2, ZFMISC_1:def 10; verum