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

theorem
  M1 is_congruent_Matrix_of M2 & n>0 implies M1@ is_congruent_Matrix_of M2@
proof
  assume that
A1: M1 is_congruent_Matrix_of M2 and
A2: n>0;
  consider M4 be Matrix of n,K such that
A3: M4 is invertible and
A4: M1=M4@*M2*M4 by A1;
A5: width (M4@)=n by MATRIX_0:24;
A6: width (M2*M4)=n & len (M2*M4)=n by MATRIX_0:24;
A7: len M2=n by MATRIX_0:24;
A8: width M2=n by MATRIX_0:24;
  take M4;
A9: len M4=n by MATRIX_0:24;
A10: width M4=n by MATRIX_0:24;
  then M4@*M2@*M4=M4@*M2@*(M4@)@ by A2,A9,MATRIX_0:57
    .=(M2*M4)@*(M4@)@ by A2,A10,A9,A8,MATRIX_3:22
    .=(M4@*(M2*M4))@ by A2,A5,A6,MATRIX_3:22
    .=M1@ by A4,A9,A8,A7,A5,MATRIX_3:33;
  hence thesis by A3;
end;
