
theorem Th43:
  for x,y,f being object holds x in the carrier of 1GateCircStr(<*x,y
  *>,f) & y in the carrier of 1GateCircStr(<*x,y*>,f) & [<*x,y*>,f] in the
  carrier of 1GateCircStr(<*x,y*>,f)
proof
  let x,y,f be object;
  set p = <*x,y*>;
  x in {x,y} by TARSKI:def 2;
  then
A1: x in rng p by FINSEQ_2:127;
  y in {x,y} by TARSKI:def 2;
  then
A2: y in rng p by FINSEQ_2:127;
  the carrier of 1GateCircStr(p,f) = rng p \/ {[p,f]} & [p,f] in {[p,f]}
  by CIRCCOMB:def 6,TARSKI:def 1;
  hence thesis by A1,A2,XBOOLE_0:def 3;
end;
