reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem Th34:
  g. [.a,b.] = [.g.a,g.b.]
proof
  thus g. [.a,b.] = g.(a" * b" * a * b) by GROUP_5:16
    .= g.(a" * b" * a) * g.b by Def6
    .= g.(a" * b") * g.a * g.b by Def6
    .= g.(a") * g.(b") * g.a * g.b by Def6
    .= (g.a)" * g.(b") * g.a * g.b by Th32
    .= (g.a)" * (g.b)" * g.a * g.b by Th32
    .= [.g.a,g.b.] by GROUP_5:16;
end;
