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;
