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

theorem
  M1 is symmetric implies M1@*M1 is symmetric
proof
  assume
A1: M1 is symmetric;
  per cases by NAT_1:3;
  suppose
A2: n>0;
A3: width M1=n & len M1=n by MATRIX_0:24;
    (M1@*M1)@=(M1*M1)@ by A1,MATRIX_6:def 5
      .=M1@*M1@ by A2,A3,MATRIX_3:22
      .=M1@*M1 by A1,MATRIX_6:def 5;
    hence thesis by MATRIX_6:def 5;
  end;
  suppose
    n=0;
    then (M1@*M1)@=M1@*M1 by MATRIX_0:45;
    hence thesis by MATRIX_6:def 5;
  end;
end;
