
theorem Th115:
  for x,y,z being set for s being State of GFA3CarryCirc(x,y,z)
for a1,a2,a3 being Element of BOOLEAN st a1 = s.x & a2 = s.y & a3 = s.z holds (
Following s).[<*x,y*>,nor2] = 'not' a1 '&' 'not' a2 & (Following s).[<*y,z*>,
  nor2] = 'not' a2 '&' 'not' a3 & (Following s).[<*z,x*>,nor2] = 'not' a3 '&'
  'not' a1
proof
  let x,y,z be set;
  set f1 = nor2, f2 = nor2, f3 = nor2;
  let s be State of GFA3CarryCirc(x,y,z);
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  let a1,a2,a3 be Element of BOOLEAN such that
A1: a1 = s.x and
A2: a2 = s.y and
A3: a3 = s.z;
  set S = GFA3CarryStr(x,y,z);
A4: InnerVertices S = the carrier' of S by FACIRC_1:37;
A5: y in the carrier of S by Th111;
A6: x in the carrier of S by Th111;
A7: dom s = the carrier of S by CIRCUIT1:3;
  xy in InnerVertices GFA3CarryStr(x,y,z) by Th112;
  hence (Following s).[<*x,y*>,f1] = f1.(s*<*x,y*>) by A4,FACIRC_1:35
    .= f1.<*a1,a2*> by A1,A2,A7,A6,A5,FINSEQ_2:125
    .= 'not' a1 '&' 'not' a2 by TWOSCOMP:54;
A8: z in the carrier of S by Th111;
  yz in InnerVertices GFA3CarryStr(x,y,z) by Th112;
  hence (Following s).[<*y,z*>,f2] = f2.(s*<*y,z*>) by A4,FACIRC_1:35
    .= f2.<*a2,a3*> by A2,A3,A7,A5,A8,FINSEQ_2:125
    .= 'not' a2 '&' 'not' a3 by TWOSCOMP:54;
  zx in InnerVertices GFA3CarryStr(x,y,z) by Th112;
  hence (Following s).[<*z,x*>,f3] = f3.(s*<*z,x*>) by A4,FACIRC_1:35
    .= f3.<*a3,a1*> by A1,A3,A7,A6,A8,FINSEQ_2:125
    .= 'not' a3 '&' 'not' a1 by TWOSCOMP:54;
end;
