reserve X,Y for set;
reserve G for Group;
reserve n for Nat;

theorem Th11:
  for X,Y being non empty set
  for p being Function of X,Y st p is bijective holds
  SymGroupsIso(p) is one-to-one
  proof
    let X,Y be non empty set;
    let p be Function of X,Y such that
A1: p is bijective;
    set h = SymGroupsIso(p);
A2: rng p = Y by A1,FUNCT_2:def 3;
    reconsider p1 = p" as Function of Y,X by A1,A2,FUNCT_2:25;
A3: id X = p1*p by A1,A2,FUNCT_2:29;
    let x,y be object such that
A4: x in dom h & y in dom h and
A5: h.x = h.y;
    reconsider x, y as Element of SymGroup(X) by A4;
    reconsider f = x, g = y as Permutation of X by Th5;
    h.x = p*f*p1 & h.y = p*g*p1 by A1,Def3;
    then p*f*(p1*p) = p*g*p1*p by A5,RELAT_1:36;
    then p*f*(p1*p) = p*g*(p1*p) by RELAT_1:36;
    then p*f = p*g*id X by A3,FUNCT_2:17;
    then p1*(p*f) = p1*(p*g) by FUNCT_2:17;
    then p1*p*f = p1*(p*g) by RELAT_1:36;
    then p1*p*f = p1*p*g by RELAT_1:36;
    then f = id X*g by A3,FUNCT_2:17;
    hence thesis by FUNCT_2:17;
  end;
