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

theorem
  M1 is invertible & M2*M1=M3*M1 implies M2=M3
proof
  assume that
A1: M1 is invertible and
A2: M2*M1=M3*M1;
A3: M1~ is_reverse_of M1 by A1,MATRIX_6:def 4;
A4: width M2=n by MATRIX_0:24;
A5: width M1=n & len M1=n by MATRIX_0:24;
A6: width M3=n by MATRIX_0:24;
A7: len (M1~)=n by MATRIX_0:24;
  M2=M2*(1.(K,n)) by MATRIX_3:19
    .=M2*(M1*M1~) by A3,MATRIX_6:def 2
    .=(M3*M1)*M1~ by A2,A5,A4,A7,MATRIX_3:33
    .=M3*(M1*M1~) by A5,A6,A7,MATRIX_3:33
    .=M3*(1.(K,n)) by A3,MATRIX_6:def 2
    .=M3 by MATRIX_3:19;
  hence thesis;
end;
