reserve x,y,c for set;

theorem Th2:
  for x,y,c being non pair set holds InputVertices BorrowStr(x,y,c)
  is without_pairs
proof
  let x,y,c be non pair set;
  set M = BorrowStr(x,y,c), MI = BorrowIStr(x,y,c);
  set S = 1GateCircStr(<*[<*x,y*>, and2a], [<*y,c*>, and2], [<*x,c*>, and2a]*>
  , or3);
  given xx being pair object such that
A1: xx in InputVertices M;
A2: 1GateCircStr(<*x,y*>, and2a) tolerates 1GateCircStr(<*y,c*>, and2) by
CIRCCOMB:47;
A3: InnerVertices 1GateCircStr(<*x,c*>, and2a) = {[<*x,c*>, and2a]} &
  1GateCircStr(<*x,y*>,and2a) +* 1GateCircStr(<*y,c*>,and2) tolerates
  1GateCircStr(<*x,c*>,and2a) by CIRCCOMB:42,47;
  InnerVertices 1GateCircStr(<*x,y*>, and2a) = {[<*x,y*>, and2a]} &
  InnerVertices 1GateCircStr(<*y,c*>, and2) = {[<*y,c*>, and2]} by CIRCCOMB:42;
  then
  InnerVertices (1GateCircStr(<*x,y*>,and2a) +* 1GateCircStr(<*y,c*>,and2
  )) = {[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]} by A2,CIRCCOMB:11;
  then
A4: InnerVertices MI = {[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]} \/ {[<*x,c*>,
  and2a]} by A3,CIRCCOMB:11
    .= {[<*x,y*>,and2a], [<*y,c*>,and2]} \/ {[<*x,c*>,and2a]} by ENUMSET1:1
    .= {[<*x,y*>,and2a], [<*y,c*>,and2], [<*x,c*>,and2a]} by ENUMSET1:3;
  InputVertices S = {[<*x,y*>,and2a], [<*y,c*>,and2], [<*x,c*>,and2a]} by
FACIRC_1:42;
  then
A5: InputVertices S \ InnerVertices MI = {} by A4,XBOOLE_1:37;
  InputVertices 1GateCircStr(<*x,y*>,and2a) is without_pairs &
  InputVertices 1GateCircStr(<*y,c*>,and2) is without_pairs by FACIRC_1:41;
  then InputVertices 1GateCircStr(<*x,c*>,and2a) is without_pairs &
  InputVertices ( 1GateCircStr(<*x,y*>,and2a)+*1GateCircStr(<*y,c*>,and2 )) is
  without_pairs by FACIRC_1:9,41;
  then
A6: InputVertices MI is without_pairs by FACIRC_1:9;
  InnerVertices S is Relation by FACIRC_1:38;
  then
  InputVertices M = (InputVertices MI) \/ (InputVertices S \ InnerVertices
  MI) by A6,FACIRC_1:6;
  hence thesis by A6,A1,A5;
end;
