reserve i,n for Nat,
  K for Field,
  M1,M2,M3,M4 for Matrix of n,K;

theorem
  M1 is Orthogonal implies M2*M1 is_congruent_Matrix_of M1*M2
proof
A1: len M2=n & width (M1~)=n by MATRIX_0:24;
  assume
A2: M1 is Orthogonal;
  then M1 is invertible by MATRIX_6:def 7;
  then
A3: M1~ is_reverse_of M1 by MATRIX_6:def 4;
  take M1;
  len M1=n & width M1=n by MATRIX_0:24;
  then M1~*(M1*M2)*M1 =((M1~*M1)*M2)*M1 by A1,MATRIX_3:33
    .=((1.(K,n))*M2)*M1 by A3,MATRIX_6:def 2
    .=M2*M1 by MATRIX_3:18;
  hence thesis by A2,MATRIX_6:def 7;
end;
