
theorem Th108:
  for x,y,z being set st x <> [<*y,z*>,nor2] & y <> [<*z,x*>,
nor2] & z <> [<*x,y*>,nor2] holds InputVertices GFA3CarryIStr(x,y,z) = {x,y,z
  }
proof
  let x,y,z be set;
  set f1 = nor2, f2 = nor2, f3 = nor2;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  set Cxy = 1GateCircStr(<*x,y*>,f1);
  set Cyz = 1GateCircStr(<*y,z*>,f2);
  set Czx = 1GateCircStr(<*z,x*>,f3);
  assume that
A1: x <> yz and
A2: y <> zx & z <> xy;
A3: not xy in {y,z} by A1,A2,Lm1;
A4: not zx in {x,y,z} by A1,A2,Lm1;
A5: y <> yz by FACIRC_2:2;
A6: ( not z in {xy, yz})& not x in {xy, yz} by A1,A2,Lm1;
A7: Cxy tolerates Cyz by CIRCCOMB:47;
  InputVertices GFA3CarryIStr(x,y,z) = (InputVertices(Cxy +* Cyz) \
  InnerVertices(Czx)) \/ (InputVertices(Czx) \ InnerVertices(Cxy +* Cyz)) by
CIRCCMB2:5,CIRCCOMB:47
    .= ((InputVertices(Cxy) \ InnerVertices(Cyz)) \/ (InputVertices(Cyz) \
  InnerVertices(Cxy))) \ InnerVertices(Czx) \/ (InputVertices(Czx) \
  InnerVertices(Cxy +* Cyz)) by CIRCCMB2:5,CIRCCOMB:47
    .= ((InputVertices(Cxy) \ InnerVertices(Cyz)) \/ (InputVertices(Cyz) \
  InnerVertices(Cxy))) \ InnerVertices(Czx) \/ (InputVertices(Czx) \ (
  InnerVertices(Cxy) \/ InnerVertices(Cyz))) by A7,CIRCCOMB:11
    .= ((InputVertices(Cxy) \ {yz}) \/ (InputVertices(Cyz) \ InnerVertices(
  Cxy))) \ InnerVertices(Czx) \/ (InputVertices(Czx) \ (InnerVertices(Cxy) \/
  InnerVertices(Cyz))) by CIRCCOMB:42
    .= ((InputVertices(Cxy) \ {yz}) \/ (InputVertices(Cyz) \ {xy})) \
  InnerVertices(Czx) \/ (InputVertices(Czx) \ (InnerVertices(Cxy) \/
  InnerVertices(Cyz))) by CIRCCOMB:42
    .= ((InputVertices(Cxy) \ {yz}) \/ (InputVertices(Cyz) \ {xy})) \ {zx}
  \/ (InputVertices(Czx) \ (InnerVertices(Cxy) \/ InnerVertices(Cyz))) by
CIRCCOMB:42
    .= ((InputVertices(Cxy) \ {yz}) \/ (InputVertices(Cyz) \ {xy})) \ {zx}
  \/ (InputVertices(Czx) \ ({xy} \/ InnerVertices(Cyz))) by CIRCCOMB:42
    .= ((InputVertices(Cxy) \ {yz}) \/ (InputVertices(Cyz) \ {xy})) \ {zx}
  \/ (InputVertices(Czx) \ ({xy} \/ {yz})) by CIRCCOMB:42
    .= (({x,y} \ {yz}) \/ (InputVertices(Cyz) \ {xy})) \ {zx} \/ (
  InputVertices(Czx) \ ({xy} \/ {yz})) by FACIRC_1:40
    .= (({x,y} \ {yz}) \/ ({y,z} \ {xy})) \ {zx} \/ (InputVertices(Czx) \ ({
  xy} \/ {yz})) by FACIRC_1:40
    .= (({x,y} \ {yz}) \/ ({y,z} \ {xy})) \ {zx} \/ ({z,x} \ ({xy} \/ {yz}))
  by FACIRC_1:40
    .= (({x,y} \ {yz}) \/ ({y,z} \ {xy})) \ {zx} \/ ({z,x} \ {xy,yz}) by
ENUMSET1:1
    .= (({x,y} \/ ({y,z} \ {xy})) \ {zx}) \/ ({z,x} \ {xy,yz}) by A1,A5,
FACIRC_2:1
    .= ({x,y} \/ {y,z}) \ {zx} \/ ({z,x} \ {xy,yz}) by A3,ZFMISC_1:57
    .= ({x,y} \/ {y,z}) \ {zx} \/ {z,x} by A6,ZFMISC_1:63
    .= {x,y,y,z} \ {zx} \/ {z,x} by ENUMSET1:5
    .= {y,y,x,z} \ {zx} \/ {z,x} by ENUMSET1:67
    .= {y,x,z} \ {zx} \/ {z,x} by ENUMSET1:31
    .= {x,y,z} \ {zx} \/ {z,x} by ENUMSET1:58
    .= {x,y,z} \/ {z,x} by A4,ZFMISC_1:57
    .= {x,y,z,z,x} by ENUMSET1:9
    .= {x,y,z,z} \/ {x} by ENUMSET1:10
    .= {z,z,x,y} \/ {x} by ENUMSET1:73
    .= {z,x,y} \/ {x} by ENUMSET1:31
    .= {z,x,y,x} by ENUMSET1:6
    .= {x,x,y,z} by ENUMSET1:70
    .= {x,y,z} by ENUMSET1:31;
  hence thesis;
end;
