
theorem Th13:
  for G being Group for H being strict Subgroup of G for x being
  Element of G holds x in carr G.H iff x in H
proof
  let G be Group;
  let H be strict Subgroup of G;
  let x be Element of G;
  thus x in carr G.H implies x in H
  proof
    assume x in carr G.H;
    then x in the carrier of H by Def1;
    hence thesis by STRUCT_0:def 5;
  end;
  assume
A1: x in H;
  carr G.H = the carrier of H by Def1;
  hence thesis by A1,STRUCT_0:def 5;
end;
