reserve x,y,z,c for object,
  f for Function of 2-tuples_on BOOLEAN, BOOLEAN;
reserve s for State of 2GatesCircuit(x,y,c,f);

theorem Th68:
  for x,y,c being non pair object holds InputVertices MajorityStr(x,y
  ,c) is without_pairs
proof
  let x,y,c be non pair object;
  set M = MajorityStr(x,y,c), MI = MajorityIStr(x,y,c);
  set S = 1GateCircStr(<*[<*x,y*>, '&'], [<*y,c*>, '&'], [<*c,x*>, '&']*>, or3
  );
  given xx being pair object such that
A1: xx in InputVertices M;
A2: 1GateCircStr(<*x,y*>, '&') tolerates 1GateCircStr(<*y,c*>, '&') by
CIRCCOMB:43;
  1GateCircStr(<*x,y*>, '&') tolerates 1GateCircStr(<*c,x*>, '&') &
1GateCircStr(<*y,c*>, '&') tolerates 1GateCircStr(<*c,x*>, '&') by CIRCCOMB:43;
  then
A3: 1GateCircStr(<*x,y*>, '&') +* 1GateCircStr(<*y,c*>, '&') tolerates
  1GateCircStr(<*c,x*>, '&') by A2,CIRCCOMB:3;
  InnerVertices 1GateCircStr(<*x,y*>, '&') = {[<*x,y*>, '&']} &
  InnerVertices 1GateCircStr(<*y,c*>, '&') = {[<*y,c*>, '&']} by CIRCCOMB:42;
  then InnerVertices 1GateCircStr(<*c,x*>, '&') = {[<*c,x*>, '&']} &
InnerVertices (1GateCircStr(<*x,y*>, '&') +* 1GateCircStr(<*y,c*>, '&')) = {[<*
  x,y*>, '&']} \/ {[<*y,c*>, '&']} by A2,CIRCCOMB:11,42;
  then
A4: InnerVertices MI = {[<*x,y*>,'&']} \/ {[<*y,c*>,'&']} \/ {[<*c,x*>, '&'
  ]} by A3,CIRCCOMB:11
    .= {[<*x,y*>,'&'], [<*y,c*>,'&']} \/ {[<*c,x*>,'&']} by ENUMSET1:1
    .= {[<*x,y*>,'&'], [<*y,c*>,'&'], [<*c,x*>,'&']} by ENUMSET1:3;
  InputVertices S = {[<*x,y*>,'&'], [<*y,c*>,'&'], [<*c,x*>,'&']} by Th42;
  then
A5: InputVertices S \ InnerVertices MI = {} by A4,XBOOLE_1:37;
  InputVertices 1GateCircStr(<*x,y*>,'&') is without_pairs & InputVertices
  1GateCircStr(<*y,c*>,'&') is without_pairs by Th41;
  then
A6: InputVertices (1GateCircStr(<*x,y*>,'&')+*1GateCircStr(<*y,c*>,'&')) is
  without_pairs by Th8,CIRCCOMB:47;
  InputVertices 1GateCircStr(<*c,x*>,'&') is without_pairs by Th41;
  then
A7: InputVertices MI is without_pairs by A6,Th8,CIRCCOMB:47;
  InnerVertices S is Relation by Th38;
  then
  InputVertices M = (InputVertices MI) \/ (InputVertices S \ InnerVertices
  MI) by A7,Th6;
  hence thesis by A7,A1,A5;
end;
