reserve x,y,z,c for object,
  f for Function of 2-tuples_on BOOLEAN, BOOLEAN;

theorem Th57:
  c <> [<*x,y*>, f] implies InputVertices 2GatesCircStr(x,y,c,f) = {x,y,c}
proof
  assume
A1: c <> [<*x,y*>, f];
  set S = 2GatesCircStr(x,y,c,f);
  set S1 = 1GateCircStr(<*x,y*>, f);
  set p = <*[<*x,y*>, f], c*>;
  set S2 = 1GateCircStr(p, f);
  set R = the ResultSort of S;
A2: the carrier of S = (the carrier of S1) \/ the carrier of S2 by
CIRCCOMB:def 2;
A3: rng <*x,y*> = {x,y} by FINSEQ_2:127;
  then
A4: the carrier of S1 = {x,y} \/ {[<*x,y*>,f]} by CIRCCOMB:def 6;
A5: rng R = {[<*x,y*>, f], 2GatesCircOutput(x,y,c,f)} by Th56
    .= {[<*x,y*>, f], [p,f]};
A6: rng p = {[<*x,y*>, f], c} by FINSEQ_2:127;
  then
A7: the carrier of S2 = {[<*x,y*>, f], c} \/ {[p,f]} by CIRCCOMB:def 6;
  thus InputVertices S c= {x,y,c}
  proof
    let z be object;
    assume
A8: z in InputVertices S;
    then z in the carrier of S1 or z in the carrier of S2 by A2,XBOOLE_0:def 3;
    then
    z in {x,y} or z in {[<*x,y*>,f]} or z in {[<*x,y*>, f], c} or z in {[
    p,f]} by A4,A7,XBOOLE_0:def 3;
    then
A9: z = x or z = y or z = [<*x,y*>,f] or z = c or z = [p,f] by TARSKI:def 1
,def 2;
    not z in rng R by A8,XBOOLE_0:def 5;
    hence thesis by A5,A9,ENUMSET1:def 1,TARSKI:def 2;
  end;
  let z be object;
   reconsider zz=z as set by TARSKI:1;
  assume z in {x,y,c};
  then
A10: z = x or z = y or z = c by ENUMSET1:def 1;
  then z in {x,y} or z in rng p by A6,TARSKI:def 2;
  then z in InputVertices S1 or z in InputVertices S2 by A3,CIRCCOMB:42;
  then
A11: z in the carrier of S by A2,XBOOLE_0:def 3;
  z in {x,y} & [<*x,y*>, f] in rng p or z in {c} by A6,A10,TARSKI:def 1,def 2;
  then
A12: the_rank_of zz in the_rank_of [<*x,y*>, f] & the_rank_of [<*x,y*>, f] in
  the_rank_of [p,f] or z = c & c in rng p by A3,A6,CLASSES1:82,TARSKI:def 1
,def 2;
  then the_rank_of zz in the_rank_of [p,f] by CLASSES1:82,ORDINAL1:10;
  then
A13: z <> [p,f];
  z <> [<*x,y*>, f] by A1,A12;
  then not z in rng R by A5,A13,TARSKI:def 2;
  hence thesis by A11,XBOOLE_0:def 5;
end;
