
theorem Th48:
  for x,y,z being set holds [<*x,y*>,and2c] in InnerVertices
GFA1CarryStr(x,y,z) & [<*y,z*>,and2a] in InnerVertices GFA1CarryStr(x,y,z) & [
<*z,x*>,and2] in InnerVertices GFA1CarryStr(x,y,z) & GFA1CarryOutput(x,y,z) in
  InnerVertices GFA1CarryStr(x,y,z)
proof
  let x,y,z be set;
  set f1 = and2c, f2 = and2a, f3 = and2;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  InnerVertices GFA1CarryStr(x,y,z) = {xy,yz,zx} \/ {GFA1CarryOutput(x,y,z
  )} by Th42
    .= {xy,yz,zx,GFA1CarryOutput(x,y,z)} by ENUMSET1:6;
  hence thesis by ENUMSET1:def 2;
end;
