reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th61:
  for F,G be Function,y st y in rng (G*F) & G is one-to-one holds
  ex x st x in dom G & x in rng F & G"{y}={x} & F"{x}=(G*F)"{y}
proof
  let F,G be Function,y such that
A1: y in rng (G*F) and
A2: G is one-to-one;
  consider x being object such that
A3: x in dom (G*F) and
A4: (G*F).x=y by A1,FUNCT_1:def 3;
A5: F.x in dom G by A3,FUNCT_1:11;
A6: G.(F.x)=y by A3,A4,FUNCT_1:12;
  then G.(F.x) in {y} by TARSKI:def 1;
  then
A7: F.x in G"{y} by A5,FUNCT_1:def 7;
A8: F"{F.x} c= (G*F)"{y}
  proof
    let d be object such that
A9: d in F"{F.x};
A10: d in dom F by A9,FUNCT_1:def 7;
    F.d in {F.x} by A9,FUNCT_1:def 7;
    then
A11: F.d = F.x by TARSKI:def 1;
    then G.(F.d) in {y} by A6,TARSKI:def 1;
    then
A12: (G*F).d in {y} by A10,FUNCT_1:13;
    d in dom (G*F) by A5,A10,A11,FUNCT_1:11;
    hence thesis by A12,FUNCT_1:def 7;
  end;
  y in rng G by A1,FUNCT_1:14;
  then consider Fx be object such that
A13: G"{y}={Fx} by A2,FUNCT_1:74;
  x in dom F by A3,FUNCT_1:11;
  then
A14: F.x in rng F by FUNCT_1:def 3;
A15: F.x in dom G by A3,FUNCT_1:11;
  (G*F)"{y} c= F"{F.x}
  proof
    let d be object such that
A16: d in (G*F)"{y};
A17: d in dom (G*F) by A16,FUNCT_1:def 7;
    then
A18: d in dom F by FUNCT_1:11;
    (G*F).d in {y} by A16,FUNCT_1:def 7;
    then
A19: G.(F.d) in {y} by A17,FUNCT_1:12;
A20: F.d in dom G by A17,FUNCT_1:11;
    F.x=Fx by A13,A7,TARSKI:def 1;
    then F.d in {F.x} by A13,A20,A19,FUNCT_1:def 7;
    hence thesis by A18,FUNCT_1:def 7;
  end;
  then
A21: F"{F.x} = (G*F)"{y} by A8;
  G"{y}={F.x} by A13,A7,TARSKI:def 1;
  hence thesis by A15,A14,A21;
end;
