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

theorem Th10:
  for X,Y being non empty set
  for p being Function of X,Y st p is bijective holds
  SymGroupsIso(p) is multiplicative
  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;
    let x,y be Element of SymGroup(X);
    reconsider p1 = p" as Function of Y,X by A1,A2,FUNCT_2:25;
A3: id X = p1*p by A2,A1,FUNCT_2:29;
A4: h.x = p*x*p1 & h.y = p*y*p1 by A1,Def3;
    reconsider f = x, g = y as Permutation of X by Th5;
    thus h.(x*y) = p*(x*y)*p1 by A1,Def3
    .= p*(g*f)*p1 by Def2
    .= p*(g*(id X)*f)*p1 by FUNCT_2:17
    .= p*(g*p1*p*f)*p1 by A3,RELAT_1:36
    .= p*(g*p1*(p*f))*p1 by RELAT_1:36
    .= p*(g*p1)*(p*f)*p1 by RELAT_1:36
    .= p*(g*p1)*(p*f*p1) by RELAT_1:36
    .= p*g*p1*(p*f*p1) by RELAT_1:36
    .= h.x * h.y by A4,Def2;
  end;
