reserve
  j, k, l, n, m, t,i for Nat,
  K for comRing, 
  a for Element of K,
  M,M1,M2 for Matrix of n,m,K,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem
  a is left_invertible & a is right_mult-cancelable &
  l in dom (1.(K,n)) & n>0 implies (SXCol((1.(K,n)),l,a))~ =
  SXCol((1.(K,n)),l,/a)
proof
  assume that
A0: a is left_invertible & a is right_mult-cancelable and
A1: l in dom (1.(K,n)) and
A3: n>0;
A4: (SXLine((1.(K,n)),l,a))~ = SXLine((1.(K,n)),l,/a) by A0,A1,Th14;
a3: SXLine((1.(K,n)),l,a) is invertible by A0,A1,Th14;
  len (1.(K,n)) = n & width (1.(K,n)) = n by MATRIX_0:24;
  then
A5: dom (1.(K,n)) = Seg width (1.(K,n)) by FINSEQ_1:def 3;
  (1.(K,n))@ = (1.(K,n)) by MATRIX_6:10;
  then SXCol((1.(K,n)),l,/a) = (SXLine((1.(K,n)),l,/a))@ by A1,A3,A5,Th16
    .= (SXLine((1.(K,n)),l,a)@)~ by A4,MATRIX_6:13,a3
    .= (SXLine((1.(K,n))@,l,a)@)~ by MATRIX_6:10
    .= (SXCol((1.(K,n)),l,a))~ by A1,A3,A5,Th16;
  hence thesis;
end;
