reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th79:
  a |^ carr (Omega).G = {b : a,b are_conjugated}
proof
  set A = a |^ carr (Omega).G;
  set B = {b : a,b are_conjugated};
  thus A c= B
  proof
    let x be object;
    assume
A1: x in A;
    then reconsider b = x as Element of G;
    ex g st x = a |^ g & g in carr(Omega).G by A1,Th42;
    then b,a are_conjugated;
    hence thesis;
  end;
  let x be object;
  assume x in B;
  then consider b such that
A2: x = b and
A3: a,b are_conjugated;
  ex g st b = a |^ g by A3,Def7;
  hence thesis by A2,Th42;
end;
