
theorem Th42:
  for f being object, p being FinSequence holds InputVertices
  1GateCircStr(p,f) = rng p & InnerVertices 1GateCircStr(p,f) = {[p,f]}
proof
  let f be object;
  let p be FinSequence;
A1: the ResultSort of 1GateCircStr(p,f) = (p,f) .--> [p,f] by Th40;
  then
A2: rng the ResultSort of 1GateCircStr(p,f) = {[p,f]} by FUNCOP_1:8;
A3: the carrier of 1GateCircStr(p,f) = (rng p) \/ {[p,f]} by Def6;
  hence InputVertices 1GateCircStr(p,f) c= rng p by A2,XBOOLE_1:43;
A4: now
    assume [p,f] in rng p;
    then consider x being object such that
A5: [x,[p,f]] in p by XTUPLE_0:def 13;
A6: {x,[p,f]} in {{x,[p,f]},{x}} by TARSKI:def 2;
A7: {p,f} in {{p,f},{p}} by TARSKI:def 2;
A8: p in {p,f} by TARSKI:def 2;
    [p,f] in {x,[p,f]} by TARSKI:def 2;
    hence contradiction by A5,A8,A7,A6,XREGULAR:9;
  end;
  thus rng p c= InputVertices 1GateCircStr(p,f)
  proof
    let x be object;
    assume
A9: x in rng p;
    then
A10: x in (rng p) \/ {[p,f]} by XBOOLE_0:def 3;
    not x in {[p,f]} by A4,A9,TARSKI:def 1;
    hence thesis by A2,A3,A10,XBOOLE_0:def 5;
  end;
  thus thesis by A1,FUNCOP_1:8;
end;
