let x1 be set ; :: thesis: for X being non empty finite set
for f being Function of (1 -tuples_on X),X
for S being Signature of X st x1 in the carrier of S & not Output (1GateCircStr (<*x1*>,f)) in InputVertices S holds
InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S
let X be non empty finite set ; :: thesis: for f being Function of (1 -tuples_on X),X
for S being Signature of X st x1 in the carrier of S & not Output (1GateCircStr (<*x1*>,f)) in InputVertices S holds
InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S
set p = <*x1*>;
let f be Function of (1 -tuples_on X),X; :: thesis: for S being Signature of X st x1 in the carrier of S & not Output (1GateCircStr (<*x1*>,f)) in InputVertices S holds
InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S
let S be Signature of X; :: thesis: ( x1 in the carrier of S & not Output (1GateCircStr (<*x1*>,f)) in InputVertices S implies InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S )
assume x1 in the carrier of S ; :: thesis: ( Output (1GateCircStr (<*x1*>,f)) in InputVertices S or InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S )
then {x1} c= the carrier of S by ZFMISC_1:31;
then rng <*x1*> c= the carrier of S by FINSEQ_1:38;
hence ( Output (1GateCircStr (<*x1*>,f)) in InputVertices S or InputVertices (S +* (1GateCircStr (<*x1*>,f))) = InputVertices S ) by Th35; :: thesis: verum