 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   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-rank free Z_Module,
   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 INT.Ring,
   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;
    A2: [:rng p1,rng p2:] c= [:INT,the carrier of V1:] by ZFMISC_1:96;
    A3: rng <:p1,p2:> c= [:rng p1,rng p2:] &
    [:INT,the carrier of V1:] = dom (the lmult of V1)
    by FUNCT_2:def 1,FUNCT_3:51;
    thus dom lmlt(p1,p2) = dom((the lmult of V1)*<:p1,p2:> ) by FUNCOP_1:def 3
    .= dom <:p1,p2:> by A2,A3,RELAT_1:27,XBOOLE_1:1
    .= dom p1 /\ dom p2 by FUNCT_3:def 7
    .= dom p1 by A1;
  end;
