
theorem Th13:
  for x,y,c being set st
  x <> [<*y,c*>, and2] & y <> [<*x,c*>, and2a] & c <> [<*x,y*>, and2a]
  holds InputVertices BorrowIStr(x,y,c) = {x,y,c}
proof
  let x,y,c be set;
  assume that
A1: x <> [<*y,c*>, and2] and
A2: y <> [<*x,c*>, and2a] and
A3: c <> [<*x,y*>, and2a];
A4: 1GateCircStr(<*x,y*>,and2a) tolerates 1GateCircStr(<*y,c*>,and2)
  by CIRCCOMB:47;
A5: y in {1,y} by TARSKI:def 2;
A6: y in {2,y} by TARSKI:def 2;
A7: {1,y} in {{1},{1,y}} by TARSKI:def 2;
A8: {2,y} in {{2},{2,y}} by TARSKI:def 2;
  <*y,c*> = <*y*>^<*c*> by FINSEQ_1:def 9;
  then
A9: <*y*> c= <*y,c*> by FINSEQ_6:10;
  <*y*> = {[1,y]} by FINSEQ_1:def 5;
  then
A10: [1,y] in <*y*> by TARSKI:def 1;
A11: <*y,c*> in {<*y,c*>} by TARSKI:def 1;
A12: {<*y,c*>} in {{<*y,c*>},{<*y,c*>,and2}} by TARSKI:def 2;
  then
A13: y <> [<*y,c*>,and2] by A5,A7,A9,A10,A11,XREGULAR:9;
A14: c in {2,c} by TARSKI:def 2;
A15: {2,c} in {{2},{2,c}} by TARSKI:def 2;
  dom<*y,c*> = Seg 2 by FINSEQ_1:89;
  then
A16: 2 in dom <*y,c*> by FINSEQ_1:1;
  <*y,c*>.2 = c;
  then [2,c] in <*y,c*> by A16,FUNCT_1:1;
  then
A17: c <> [<*y,c*>,and2] by A11,A12,A14,A15,XREGULAR:9;
  dom<*x,y*> = Seg 2 by FINSEQ_1:89;
  then
A18: 2 in dom <*x,y*> by FINSEQ_1:1;
  <*x,y*>.2 = y;
  then
A19: [2,y] in <*x,y*> by A18,FUNCT_1:1;
A20: <*x,y*> in {<*x,y*>} by TARSKI:def 1;
  {<*x,y*>} in {{<*x,y*>},{<*x,y*>,and2a}} by TARSKI:def 2;
  then y <> [<*x,y*>,and2a] by A6,A8,A19,A20,XREGULAR:9;
  then
A21: not [<*x,y*>,and2a] in {y,c} by A3,TARSKI:def 2;
A22: x in {1,x} by TARSKI:def 2;
A23: {1,x} in {{1},{1,x}} by TARSKI:def 2;
  <*x,y*> = <*x*>^<*y*> by FINSEQ_1:def 9;
  then
A24: <*x*> c= <*x,y*> by FINSEQ_6:10;
  <*x*> = {[1,x]} by FINSEQ_1:def 5;
  then
A25: [1,x] in <*x*> by TARSKI:def 1;
A26: <*x,y*> in {<*x,y*>} by TARSKI:def 1;
  {<*x,y*>} in {{<*x,y*>},{<*x,y*>,and2a}} by TARSKI:def 2;
  then
A27: x <> [<*x,y*>,and2a] by A22,A23,A24,A25,A26,XREGULAR:9;
A28: not c in {[<*x,y*>,and2a], [<*y,c*>,and2]} by A3,A17,TARSKI:def 2;
A29: not x in {[<*x,y*>,and2a], [<*y,c*>,and2]} by A1,A27,TARSKI:def 2;
A30: c in {2,c} by TARSKI:def 2;
A31: {2,c} in {{2},{2,c}} by TARSKI:def 2;
  dom<*x,c*> = Seg 2 by FINSEQ_1:89;
  then
A32: 2 in dom <*x,c*> by FINSEQ_1:1;
  <*x,c*>.2 = c;
  then
A33: [2,c] in <*x,c*> by A32,FUNCT_1:1;
A34: <*x,c*> in {<*x,c*>} by TARSKI:def 1;
  {<*x,c*>} in {{<*x,c*>},{<*x,c*>,and2a}} by TARSKI:def 2;
  then
A35: c <> [<*x,c*>,and2a] by A30,A31,A33,A34,XREGULAR:9;
A36: x in {1,x} by TARSKI:def 2;
A37: {1,x} in {{1},{1,x}} by TARSKI:def 2;
  <*x,c*> = <*x*>^<*c*> by FINSEQ_1:def 9;
  then
A38: <*x*> c= <*x,c*> by FINSEQ_6:10;
  <*x*> = {[1,x]} by FINSEQ_1:def 5;
  then
A39: [1,x] in <*x*> by TARSKI:def 1;
A40: <*x,c*> in {<*x,c*>} by TARSKI:def 1;
  {<*x,c*>} in {{<*x,c*>},{<*x,c*>,and2a}} by TARSKI:def 2;
  then x <> [<*x,c*>,and2a] by A36,A37,A38,A39,A40,XREGULAR:9;
  then
A41: not [<*x,c*>,and2a] in {x,y,c} by A2,A35,ENUMSET1:def 1;
  InputVertices BorrowIStr(x,y,c)
  = (InputVertices(1GateCircStr(<*x,y*>,and2a) +*
  1GateCircStr(<*y,c*>,and2)) \ InnerVertices(1GateCircStr(<*x,c*>,and2a))) \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  InnerVertices(1GateCircStr(<*x,y*>,and2a) +*
  1GateCircStr(<*y,c*>,and2))) by CIRCCMB2:5,CIRCCOMB:47
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \
  InnerVertices(1GateCircStr(<*y,c*>,and2))) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  InnerVertices(1GateCircStr(<*x,y*>,and2a)))) \
  InnerVertices(1GateCircStr(<*x,c*>,and2a)) \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  InnerVertices(1GateCircStr(<*x,y*>,and2a) +*
  1GateCircStr(<*y,c*>,and2))) by CIRCCMB2:5,CIRCCOMB:47
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \
  InnerVertices(1GateCircStr(<*y,c*>,and2))) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  InnerVertices(1GateCircStr(<*x,y*>,and2a)))) \
  InnerVertices(1GateCircStr(<*x,c*>,and2a)) \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  (InnerVertices(1GateCircStr(<*x,y*>,and2a)) \/
  InnerVertices(1GateCircStr(<*y,c*>,and2)))) by A4,CIRCCOMB:11
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \ {[<*y,c*>,and2]}) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  InnerVertices(1GateCircStr(<*x,y*>,and2a)))) \
  InnerVertices(1GateCircStr(<*x,c*>,and2a)) \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  (InnerVertices(1GateCircStr(<*x,y*>,and2a)) \/
  InnerVertices(1GateCircStr(<*y,c*>,and2)))) by CIRCCOMB:42
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \ {[<*y,c*>,and2]}) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \ {[<*x,y*>,and2a]})) \
  InnerVertices(1GateCircStr(<*x,c*>,and2a)) \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  (InnerVertices(1GateCircStr(<*x,y*>,and2a)) \/
  InnerVertices(1GateCircStr(<*y,c*>,and2)))) by CIRCCOMB:42
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \ {[<*y,c*>,and2]}) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]} \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  (InnerVertices(1GateCircStr(<*x,y*>,and2a)) \/
  InnerVertices(1GateCircStr(<*y,c*>,and2)))) by CIRCCOMB:42
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \ {[<*y,c*>,and2]}) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]} \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \ ({[<*x,y*>,and2a]} \/
  InnerVertices(1GateCircStr(<*y,c*>,and2)))) by CIRCCOMB:42
    .= ((InputVertices(1GateCircStr(<*x,y*>,and2a)) \ {[<*y,c*>,and2]}) \/
  (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]} \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  ({[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]})) by CIRCCOMB:42
    .= (({x,y} \
  {[<*y,c*>,and2]}) \/ (InputVertices(1GateCircStr(<*y,c*>,and2)) \
  {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]} \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  ({[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]})) by FACIRC_1:40
    .=(({x,y} \ {[<*y,c*>,and2]}) \/ ({y,c} \
  {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]} \/
  (InputVertices(1GateCircStr(<*x,c*>,and2a)) \
  ({[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]})) by FACIRC_1:40
    .=(({x,y} \ {[<*y,c*>,and2]}) \/ ({y,c} \ {[<*x,y*>,and2a]})) \
  {[<*x,c*>,and2a]} \/
  ({x,c} \ ({[<*x,y*>,and2a]} \/ {[<*y,c*>,and2]})) by FACIRC_1:40
    .= (({x,y} \ {[<*y,c*>,and2]}) \/ ({y,c} \ {[<*x,y*>,and2a]})) \
  {[<*x,c*>,and2a]} \/
  ({x,c} \ {[<*x,y*>,and2a],[<*y,c*>,and2]}) by ENUMSET1:1
    .= (({x,y} \/ ({y,c} \ {[<*x,y*>,and2a]})) \ {[<*x,c*>,and2a]}) \/
  ({x,c} \ {[<*x,y*>,and2a],[<*y,c*>,and2]}) by A1,A13,FACIRC_2:1
    .= ({x,y} \/ {y,c}) \ {[<*x,c*>,and2a]} \/
  ({x,c} \ {[<*x,y*>,and2a],[<*y,c*>,and2]}) by A21,ZFMISC_1:57
    .= ({x,y} \/ {y,c}) \ {[<*x,c*>,and2a]} \/ {x,c} by A28,A29,ZFMISC_1:63
    .= {x,y,y,c} \ {[<*x,c*>,and2a]} \/ {x,c} by ENUMSET1:5
    .= {y,y,x,c} \ {[<*x,c*>,and2a]} \/ {x,c} by ENUMSET1:67
    .= {y,x,c} \ {[<*x,c*>,and2a]} \/ {x,c} by ENUMSET1:31
    .= {x,y,c} \ {[<*x,c*>,and2a]} \/ {x,c} by ENUMSET1:58
    .= {x,y,c} \/ {x,c} by A41,ZFMISC_1:57
    .= {x,y,c,c,x} by ENUMSET1:9
    .= {x,y,c,c} \/ {x} by ENUMSET1:10
    .= {c,c,x,y} \/ {x} by ENUMSET1:73
    .= {c,x,y} \/ {x} by ENUMSET1:31
    .= {c,x,y,x} by ENUMSET1:6
    .= {x,x,y,c} by ENUMSET1:70
    .= {x,y,c} by ENUMSET1:31;
  hence thesis;
end;
