reserve x,y,c for set;

theorem Th9:
  for x,y,c being non pair set holds InputVertices BorrowStr(x,y,c)
= {x,y,c} & InnerVertices BorrowStr(x,y,c) = {[<*x,y*>,and2a], [<*y,c*>,and2],
  [<*x,c*>,and2a]} \/ {BorrowOutput(x,y,c)}
proof
  let x,y,c be non pair set;
  set xy = <*x,y*>, yc = <*y,c*>, xc = <*x,c*>;
  set xy1 =[xy,and2a], yc1 = [yc,and2], xc1 = [xc,and2a];
  set MI = BorrowIStr(x,y,c), S = 1GateCircStr(<*xy1, yc1, xc1*>, or3);
  set M = BorrowStr(x,y,c);
A1: InputVertices 1GateCircStr(<*x,y*>,and2a) = {x,y} & InputVertices
  1GateCircStr(<*x,c*>,and2a) = {x,c} by FACIRC_1:40;
A2: InputVertices 1GateCircStr(<*y,c*>,and2 ) = {y,c} by FACIRC_1:40;
A3: InnerVertices 1GateCircStr(<*x,y*>,and2a) = {[<*x,y*>,and2a]} &
  InnerVertices 1GateCircStr(<*y,c*>,and2) = {[<*y,c*>,and2]} by CIRCCOMB:42;
A4: InnerVertices S is Relation by FACIRC_1:38;
A5: InputVertices 1GateCircStr(<*x,y*>,and2a) is without_pairs &
  InputVertices 1GateCircStr(<*y,c*>,and2) is without_pairs by FACIRC_1:41;
  then
A6: 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 InputVertices MI is without_pairs by FACIRC_1:9;
  then
A7: InputVertices M = (InputVertices MI) \/ (InputVertices S \ InnerVertices
  MI) by A4,FACIRC_1:6;
A8: InnerVertices 1GateCircStr(<*x,c*>,and2a)= {[<*x,c*>,and2a]} by CIRCCOMB:42
;
  1GateCircStr(<*x,y*>,and2a) tolerates 1GateCircStr(<*y,c*>,and2) by
CIRCCOMB:47;
  then
A9: InnerVertices (1GateCircStr(<*x,y*>,and2a) +* 1GateCircStr(<*y,c*>,and2
  )) = {[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]} by A3,CIRCCOMB:11;
  1GateCircStr(<*x,y*>,and2a) +* 1GateCircStr(<*y,c*>,and2) tolerates
  1GateCircStr(<*x,c*>,and2a) by CIRCCOMB:47;
  then
A10: InnerVertices MI = {[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]} \/ {[<*x,c*>,
  and2a]} by A8,A9,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
A11: InputVertices S \ InnerVertices MI = {} by A10,XBOOLE_1:37;
  InnerVertices (1GateCircStr(<*x,y*>,and2a) +* 1GateCircStr(<*y,c*> ,
  and2)) = {[<*x,y*>,and2a], [<*y,c*>,and2]} by A9,ENUMSET1:1;
  hence InputVertices M = (InputVertices(1GateCircStr(<*x,y*>,and2a)+*
1GateCircStr(<*y,c*>,and2))) \/ InputVertices 1GateCircStr(<*x,c*>,and2a) by A6
,A7,A8,A11,FACIRC_1:7
    .= (InputVertices 1GateCircStr(<*x,y*>,and2a)) \/ (InputVertices
  1GateCircStr(<*y,c*>,and2)) \/ (InputVertices 1GateCircStr(<*x,c*>,and2a))
by A5,A3,FACIRC_1:7
    .= {x,y,y,c} \/ {c,x} by A1,A2,ENUMSET1:5
    .= {y,y,x,c} \/ {c,x} by ENUMSET1:67
    .= {y,x,c} \/ {c,x} by ENUMSET1:31
    .= {x,y,c} \/ {c,x} by ENUMSET1:58
    .= {x,y,c} \/ ({c}\/{x}) by ENUMSET1:1
    .= {x,y,c} \/ {c} \/ {x} by XBOOLE_1:4
    .= {c,x,y} \/ {c} \/ {x} by ENUMSET1:59
    .= {c,c,x,y} \/ {x} by ENUMSET1:4
    .= {c,x,y} \/ {x} by ENUMSET1:31
    .= {x,y,c} \/ {x} by ENUMSET1:59
    .= {x,x,y,c} by ENUMSET1:4
    .= {x,y,c} by ENUMSET1:31;
  MI tolerates S by CIRCCOMB:47;
  hence InnerVertices M = (InnerVertices MI) \/ (InnerVertices S) by
CIRCCOMB:11
    .= {[<*x,y*>,and2a], [<*y,c*>,and2], [<*x,c*>,and2a]} \/ {BorrowOutput(x
  ,y,c)} by A10,CIRCCOMB:42;
end;
