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 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.K by VECTSP_6:2;
      not v in Carrier KL1 by A11,XBOOLE_0:def 3;
      hence KL1.v = KL2.v by A12,VECTSP_6:2;
    end;
    suppose
      v in Carrier KL1 \/ Carrier KL2;
      then consider x being 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,FINSEQ_3:23;
      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;
