reserve n for Nat;
reserve K for Field;
reserve a,b,c,d,e,f,g,h,i,a1,b1,c1,d1,e1,f1,g1,h1,i1 for Element of K;
reserve M,N for Matrix of 3,K;
reserve p for FinSequence of REAL;

theorem Th07:
  for M,N being Matrix of 3,F_Real st
  N is symmetric holds M@ * N * M is symmetric
  proof
    let M,N be Matrix of 3,F_Real;
    assume
A1: N is symmetric;
A2: len N = 3 & width N = 3 & len M = 3 & width M = 3 by MATRIX_0:24;
    then
A3: width (M@) = len N & width N = len M by MATRIX_0:29;
    width (M@) = len N & width N <> 0 by A2,MATRIX_0:29;
    then
A4: (M@ * N)@ = (N@) * (M@@) by MATRIX_3:22
             .= (N@) * M by A2,MATRIX_0:57;
    width (M@ * N) = len M & width M <> 0 by MATRIX_0:24,A2;
    then ((M@ * N) * M)@ = (M@) * (M@ * N)@ by MATRIX_3:22
                        .= (M@) * (N * M) by A4,A1,MATRIX_6:def 5
                        .= (M@ * N) * M by A3,MATRIX_3:33;
    hence thesis by MATRIX_6:def 5;
  end;
