
theorem Th8:
  for n be Element of NAT for f be Element of IsomGroup n for g be
  Function of RLMSpace n,RLMSpace n st f = g holds f" = g"
proof
  let n be Element of NAT;
  let f be Element of IsomGroup n;
  let g be Function of RLMSpace n,RLMSpace n;
  f in the carrier of IsomGroup n;
  then
A1: f in ISOM RLMSpace n by Def9;
  then reconsider f1=f as onto isometric Function of RLMSpace n,RLMSpace n
  by Def4;
  assume
A2: f = g;
  then f1 = g;
  then
A3: g" in ISOM RLMSpace n by Def4;
  then reconsider g1 = g" as Element of IsomGroup n by Def9;
A4: rng f1 = [#](RLMSpace n) by FUNCT_2:def 3;
A5: g1 * f = g"*f1 by A1,A3,Def9
    .= id dom f1 by A2,A4,TOPS_2:52
    .= id(RLMSpace n) by FUNCT_2:def 1
    .= 1_IsomGroup n by Th7;
  f * g1 = f1*g" by A1,A3,Def9
    .= id(RLMSpace n) by A2,A4,TOPS_2:52
    .= 1_IsomGroup n by Th7;
  hence thesis by A5,GROUP_1:5;
end;
