
theorem Th47:
  for x,y,z being set holds x in the carrier of GFA1CarryStr(x,y,z
) & y in the carrier of GFA1CarryStr(x,y,z) & z in the carrier of GFA1CarryStr(
x,y,z) & [<*x,y*>,and2c] in the carrier of GFA1CarryStr(x,y,z) & [<*y,z*>,and2a
  ] in the carrier of GFA1CarryStr(x,y,z) & [<*z,x*>,and2] in the carrier of
GFA1CarryStr(x,y,z) & [<*[<*x,y*>,and2c],[<*y,z*>,and2a],[<*z,x*>,and2]*>,or3]
  in the carrier of GFA1CarryStr(x,y,z)
proof
  let x,y,z be set;
  set f1 = and2c, f2 = and2a, f3 = and2, f4 = or3;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  set xyz = [<*xy, yz, zx*>,f4];
  set Cxy = 1GateCircStr(<*x,y*>,f1);
  set Cyz = 1GateCircStr(<*y,z*>,f2);
  set Czx = 1GateCircStr(<*z,x*>,f3);
  set Cxyz = 1GateCircStr(<*xy, yz, zx*>,f4);
  set p = <*xy,yz,zx*>;
  z in the carrier of Czx by FACIRC_1:43;
  then
A1: z in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  zx in the carrier of Czx by FACIRC_1:43;
  then
A2: zx in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  y in the carrier of Cxy by FACIRC_1:43;
  then y in the carrier of Cxy +* Cyz by FACIRC_1:20;
  then
A3: y in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  the carrier of Cxyz = rng p \/ {xyz} & [p,f4] in {[p,f4]} by CIRCCOMB:def 6
,TARSKI:def 1;
  then
A4: xyz in the carrier of Cxyz by XBOOLE_0:def 3;
  yz in the carrier of Cyz by FACIRC_1:43;
  then yz in the carrier of Cxy +* Cyz by FACIRC_1:20;
  then
A5: yz in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  xy in the carrier of Cxy by FACIRC_1:43;
  then xy in the carrier of Cxy +* Cyz by FACIRC_1:20;
  then
A6: xy in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  x in the carrier of Czx by FACIRC_1:43;
  then x in the carrier of GFA1CarryIStr(x,y,z) by FACIRC_1:20;
  hence thesis by A3,A1,A6,A5,A2,A4,FACIRC_1:20;
end;
