let a be set ; :: according to FACIRC_1:def 3 :: thesis: ( a in dom <*x,y*> implies not <*x,y*> . a is pair )

assume a in dom <*x,y*> ; :: thesis: not <*x,y*> . a is pair

then <*x,y*> . a in rng <*x,y*> by FUNCT_1:def 3;

then <*x,y*> . a in {x,y} by FINSEQ_2:127;

hence not <*x,y*> . a is pair by TARSKI:def 2; :: thesis: verum

assume a in dom <*x,y*> ; :: thesis: not <*x,y*> . a is pair

then <*x,y*> . a in rng <*x,y*> by FUNCT_1:def 3;

then <*x,y*> . a in {x,y} by FINSEQ_2:127;

hence not <*x,y*> . a is pair by TARSKI:def 2; :: thesis: verum