reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem
  for M be Matrix of len b1,len b2,K holds 
  rank Mx2Tran(M,b1,b2) = the_rank_of M
proof
  let M be Matrix of len b1,len b2,K;
  thus rank Mx2Tran(M,b1,b2) = the_rank_of AutMt(Mx2Tran(M,b1,b2),b1,b2) by
Th48
    .= the_rank_of M by Th36;
end;
