 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 Th34:
  v1|--b2 = v2|--b2 implies v1 = v2
  proof
    consider KL1 be Linear_Combination of V2 such that
    A1: v1 = Sum(KL1) and
    A2: Carrier KL1 c= rng b2 and
    A3: for t st 1<=t & t<=len (v1|--b2) holds (v1|--b2)/.t = KL1.(b2/.t)
    by Def7;
    consider KL2 be Linear_Combination of V2 such that
    A4: v2 = Sum(KL2) and
    A5: Carrier KL2 c= rng b2 and
    A6: for t st 1<=t & t<=len (v2|--b2) holds (v2|--b2)/.t = KL2.(b2/.t)
    by Def7;
    assume
    A7: v1|--b2 = v2|--b2;
    A8:
    now
      let t be Nat;
      assume
      A9: 1 <= t & t <= len (v1|--b2);
      hence KL1.(b2/.t) = (v2|--b2)/.t by A7,A3
      .= KL2.(b2/.t) by A7,A6,A9;
    end;
    A10: Carrier KL1 \/ Carrier KL2 c= rng b2 by A2,A5,XBOOLE_1:8;
    now
      let v be Vector of V2;
      per cases;
      suppose A11: not v in Carrier KL1 \/ Carrier KL2;
        then not v in Carrier KL2 by XBOOLE_0:def 3;
        then
        A12: KL2.v = 0;
        not v in Carrier KL1 by A11,XBOOLE_0:def 3;
        hence KL1.v = KL2.v by A12;
      end;
      suppose v in Carrier KL1 \/ Carrier KL2;
        then consider x be object such that
        A13: x in dom b2 and
        A14: v = b2.x by A10,FUNCT_1:def 3;
        reconsider x as Nat by A13;
        x <= len b2 by A13,FINSEQ_3:25;
        then
        A15: x <= len (v1|--b2) by Def7;
        v = b2/.x & 1 <= x by A13,A14,FINSEQ_3:25,PARTFUN1:def 6;
        hence KL1.v = KL2.v by A8,A15;
      end;
    end;
    hence thesis by A1,A4,VECTSP_6:def 7;
  end;
