
theorem Th7:
  for n be Element of NAT holds 1_IsomGroup n = id (RLMSpace n)
proof
  let n be Element of NAT;
A1: id(RLMSpace n) in ISOM RLMSpace n by Def4;
  then reconsider e = id(RLMSpace n) as Element of IsomGroup n by Def9;
  now
    let h be Element of IsomGroup n;
    h in the carrier of IsomGroup n;
    then
A2: h in ISOM RLMSpace n by Def9;
    then reconsider h1=h as Function of RLMSpace n,RLMSpace n by Def4;
    thus h * e = h1*id(the carrier of RLMSpace n) by A1,A2,Def9
      .= h by FUNCT_2:17;
    thus e * h = id(the carrier of RLMSpace n)*h1 by A1,A2,Def9
      .= h by FUNCT_2:17;
  end;
  hence thesis by GROUP_1:4;
end;
