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 Th9:
  len p1 = len R1 & len p2 = len R2 implies lmlt(p1^p2,R1^R2) =
  lmlt(p1,R1)^lmlt(p2,R2)
proof
  assume that
A1: len p1=len R1 and
A2: len p2=len R2;
  reconsider r2=R2 as Element of (len p2)-tuples_on the carrier of V1 by A2,
FINSEQ_2:92;
  reconsider r1=R1 as Element of (len p1)-tuples_on the carrier of V1 by A1,
FINSEQ_2:92;
  reconsider P1=p1 as Element of (len p1)-tuples_on the carrier of K by
FINSEQ_2:92;
  reconsider P2=p2 as Element of (len p2)-tuples_on the carrier of K by
FINSEQ_2:92;
  thus lmlt(p1^p2,R1^R2) = ((the lmult of V1).:(P1,r1))^((the lmult of V1).:(
  P2,r2)) by FINSEQOP:11
    .= lmlt(p1,R1)^lmlt(p2,R2);
end;
