reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th30:
  for A,B being Matrix of n,K holds (A*B)@=(B@)*(A@)
proof
  let A,B be Matrix of n,K;
  per cases;
  suppose
A1: n<>0;
A2: width B=n by MATRIX_0:24;
    len B=n & width A=n by MATRIX_0:24;
    hence thesis by A1,A2,MATRIX_3:22;
  end;
  suppose
A3: n=0;
    hence (A*B)@=1.(K,n) by Th15
      .=(B@)*(A@) by A3,Th15;
  end;
end;
