reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

theorem Th12:
  dom p1 = dom p2 implies dom lmlt(p1,p2) = dom p1
proof
  assume
A1: dom p1 = dom p2;
  rng p1 c= the carrier of K & rng p2 c= the carrier of V1 by FINSEQ_1:def 4;
  then
A2: [:rng p1,rng p2:] c= [:the carrier of K,the carrier of V1:] by ZFMISC_1:96;
  rng <:p1,p2:> c= [:rng p1,rng p2:] & [:the carrier of K,the carrier of
  V1:] = dom (the lmult of V1) by FUNCT_2:def 1,FUNCT_3:51;
  hence dom lmlt(p1,p2) = dom <:p1,p2:> by A2,RELAT_1:27,XBOOLE_1:1
    .= dom p1 /\ dom p2 by FUNCT_3:def 7
    .= dom p1 by A1;
end;
