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 M2 is_congruent_Matrix_of M1
proof
  assume that
A1: M1 is_congruent_Matrix_of M2 and
A2: n>0;
  consider M be Matrix of n,K such that
A3: M is invertible and
A4: M1=M@*M2*M by A1;
A5: M~ is_reverse_of M by A3,MATRIX_6:def 4;
  take M~;
A6: width (M~)=n by MATRIX_0:24;
A7: len (M@)=n & width (M@)=n by MATRIX_0:24;
A8: width (M)=n by MATRIX_0:24;
A9: len (M~)=n by MATRIX_0:24;
  len (M)=n & width (M@*M2)=n by MATRIX_0:24;
  then
A10: M1*M~=(M@*M2)*(M*M~) by A4,A8,A9,MATRIX_3:33
    .=(M@*M2)*(1.(K,n)) by A5,MATRIX_6:def 2
    .=M@*M2 by MATRIX_3:19;
A11: len M2=n by MATRIX_0:24;
A12: width (M~@)=n by MATRIX_0:24;
  len M1=n & width M1=n by MATRIX_0:24;
  then M~@*M1*M~=M~@*(M1*M~) by A9,A12,MATRIX_3:33
    .=(M~@*M@)*M2 by A7,A11,A12,A10,MATRIX_3:33
    .=(M*M~)@*M2 by A2,A8,A9,A6,MATRIX_3:22
    .=(1.(K,n))@*M2 by A5,MATRIX_6:def 2
    .=(1.(K,n))*M2 by MATRIX_6:10
    .=M2 by MATRIX_3:18;
  hence thesis by A3;
end;
