
theorem Th82:
  for x,y,z being non pair set holds InputVertices GFA2CarryStr(x,
  y,z) = {x,y,z}
proof
  let x,y,z be non pair set;
  set f1 = and2a, f2 = and2c, f3 = nor2, f4 = nor3;
  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);
  set M = GFA2CarryStr(x,y,z);
  set MI = GFA2CarryIStr(x,y,z);
  set S = 1GateCircStr(<*xy,yz,zx*>,f4);
A1: InputVertices Cxy = {x,y} & InputVertices Cyz = {y,z} by FACIRC_1:40;
A2: InputVertices Czx = {z,x} by FACIRC_1:40;
A3: InputVertices S = {xy, yz, zx} by FACIRC_1:42;
A4: InnerVertices S is Relation by FACIRC_1:38;
A5: InnerVertices Cxy = {xy} & InnerVertices Cyz = {yz} by CIRCCOMB:42;
  Cxy tolerates Cyz by CIRCCOMB:47;
  then
A6: InnerVertices Czx = {zx} & InnerVertices (Cxy +* Cyz) = {xy} \/ {yz} by A5,
CIRCCOMB:11,42;
  Cxy +* Cyz tolerates Czx by CIRCCOMB:47;
  then InnerVertices MI = {xy} \/ {yz} \/ {zx} by A6,CIRCCOMB:11
    .= {xy, yz} \/ {zx} by ENUMSET1:1
    .= {xy, yz, zx} by ENUMSET1:3;
  then
A7: InputVertices S \ InnerVertices MI = {} by A3,XBOOLE_1:37;
A8: InputVertices Cxy is without_pairs & InputVertices Cyz is without_pairs
  by FACIRC_1:41;
  then
A9: InputVertices Czx is without_pairs & InputVertices (Cxy+*Cyz) is
  without_pairs by FACIRC_1:9,41;
  then InputVertices MI is without_pairs by FACIRC_1:9;
  then
  InputVertices M = (InputVertices MI) \/ (InputVertices S \ InnerVertices
  MI) by A4,FACIRC_1:6;
  hence
  InputVertices M = (InputVertices(Cxy+*Cyz)) \/ InputVertices Czx by A9,A6,A7,
FACIRC_1:7
    .= (InputVertices Cxy) \/ (InputVertices Cyz) \/ (InputVertices Czx) by A8
,A5,FACIRC_1:7
    .= {x,y,y,z} \/ {z,x} by A1,A2,ENUMSET1:5
    .= {y,y,x,z} \/ {z,x} by ENUMSET1:67
    .= {y,x,z} \/ {z,x} by ENUMSET1:31
    .= {x,y,z} \/ {z,x} by ENUMSET1:58
    .= {x,y,z} \/ ({z}\/{x}) by ENUMSET1:1
    .= {x,y,z} \/ {z} \/ {x} by XBOOLE_1:4
    .= {z,x,y} \/ {z} \/ {x} by ENUMSET1:59
    .= {z,z,x,y} \/ {x} by ENUMSET1:4
    .= {z,x,y} \/ {x} by ENUMSET1:31
    .= {x,y,z} \/ {x} by ENUMSET1:59
    .= {x,x,y,z} by ENUMSET1:4
    .= {x,y,z} by ENUMSET1:31;
end;
