 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 PROJ5:
  for V being free Z_Module, X being Basis of V, v being Vector of V
  st v in X & v <> 0.V holds Coordinate(v,X).v = 1
  proof
    let V be free Z_Module,
    X be Basis of V,
    v be Vector of V;
    assume AS: v in X & v <> 0.V;
    set f = Coordinate(v,X);
    consider KL be Linear_Combination of X such that
    A1: v = Sum KL & f.v = KL.v by defCoord;
    A3: Carrier KL c= X by VECTSP_6:def 4;
    v in {v} by TARSKI:def 1;
    then v in Lin{v} by ZMODUL02:65;
    then consider Lb be Linear_Combination of {v} such that
    A4: v = Sum Lb by ZMODUL02:64;
    A7: Carrier Lb c= {v} by VECTSP_6:def 4;
    {v} c= X by AS,ZFMISC_1:31;
    then Carrier Lb c= X by A7;
    then
    A9: Lb = KL by A4,A1,A3,Th5,VECTSP_7:def 3;
    Lb.v * v = v by A4,ZMODUL02:21
    .= (1.INT.Ring)*v;
    hence f.v = 1 by AS,A1,A9,ZMODUL01:11;
  end;
