let x1, x2 be set ; :: thesis: for X being non empty finite set
for f being Function of (2 -tuples_on X),X
for S being Signature of X st x1 in the carrier of S & x2 in the carrier of S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S
let X be non empty finite set ; :: thesis: for f being Function of (2 -tuples_on X),X
for S being Signature of X st x1 in the carrier of S & x2 in the carrier of S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S
let f be Function of (2 -tuples_on X),X; :: thesis: for S being Signature of X st x1 in the carrier of S & x2 in the carrier of S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S
A1: rng <*x1,x2*> = {x1,x2} by FINSEQ_2:147;
let S be Signature of X; :: thesis: ( x1 in the carrier of S & x2 in the carrier of S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S implies InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S )
assume ( x1 in the carrier of S & x2 in the carrier of S ) ; :: thesis: ( Output (1GateCircStr <*x1,x2*>,f) in InputVertices S or InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S )
then rng <*x1,x2*> c= the carrier of S by A1, ZFMISC_1:38;
hence ( Output (1GateCircStr <*x1,x2*>,f) in InputVertices S or InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S ) by Th39; :: thesis: verum