reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th37:
  for R being Ring, M1,M2 being Matrix of n,R
  holds M1 is invertible & M2 is invertible implies M1*M2 is invertible
  & (M1*M2)~=(M2~)*(M1~)
proof
  let R be Ring;
  let M1,M2 be Matrix of n,R;
  assume that
A1: M1 is invertible and
A2: M2 is invertible;
A3: M2~ is_reverse_of M2 by A2,Def4;
A4: M1~ is_reverse_of M1 by A1,Def4;
A5: len (M2~)=n by MATRIX_0:24;
A6: width (M1~)=n by MATRIX_0:24;
A7: len M1=n by MATRIX_0:24;
A8: width M2=n by MATRIX_0:24;
A9: width M1=n & len M2=n by MATRIX_0:24;
A10: width (M2~)=n & len (M1~)=n by MATRIX_0:24;
  width (M1*M2)=n by MATRIX_0:24;
  then
A11: (M1*M2)*(M2~*M1~)=((M1*M2)*M2~)*M1~ by A5,A10,MATRIX_3:33
    .=(M1*(M2*M2~))*M1~ by A9,A8,A5,MATRIX_3:33
    .=(M1*(1.(R,n)))*M1~ by A3
    .=M1*M1~ by MATRIX_3:19
    .=1.(R,n) by A4;
  width (M2~*M1~)=n by MATRIX_0:24;
  then (M2~*M1~)*(M1*M2)=(M2~*M1~)*M1*M2 by A9,A7,MATRIX_3:33
    .=(M2~*(M1~*M1))*M2 by A7,A6,A10,MATRIX_3:33
    .=(M2~*(1.(R,n)))*M2 by A4
    .=M2~*M2 by MATRIX_3:19
    .=1.(R,n) by A3;
  then
A12: (M2~)*(M1~) is_reverse_of M1*M2 by A11;
  then M1*M2 is invertible;
  hence thesis by A12,Def4;
end;
