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

theorem
  M2 is invertible & M1 is_similar_to M2 implies M1~ is_similar_to M2~
proof
  assume that
A1: M2 is invertible and
A2: M1 is_similar_to M2;
  consider M4 be Matrix of n,K such that
A3: M4 is invertible and
A4: M1=M4~*M2*M4 by A2;
A5: M4~ is invertible & M4~~=M4 by A3,MATRIX_6:16;
  take M4;
A6: width (M4~)=n & len M4=n by MATRIX_0:24;
A7: len M2=n & width M2=n by MATRIX_0:24;
  M2*M4 is invertible & M4~*M2~=(M2*M4)~ by A1,A3,MATRIX_6:36;
  then M4~*M2~*M4=(M4~*(M2*M4))~ by A5,MATRIX_6:36
    .=M1~ by A4,A6,A7,MATRIX_3:33;
  hence thesis by A3;
end;
