 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 LMThMBF1Y:
  for V1, V2 being finite-rank free Z_Module,
  b1 being OrdBasis of V1,
  f being bilinear-Form of V1, V2,
  v1 being Vector of V1,
  v2 being Vector of V2,
  X,Y being FinSequence of INT.Ring
  st len X = len b1 & len Y = len b1 &
  ( for k being Nat st k in Seg len b1 holds Y.k = f.(b1/.k, v2) ) &
  X = (v1 |-- b1)
  holds X "*" Y = f.(v1, v2)
  proof
    let V1, V2 be finite-rank free Z_Module,
    b1 be OrdBasis of V1,
    f be bilinear-Form of V1, V2,
    v1 be Vector of V1,
    v2 be Vector of V2,
    X,Y be FinSequence of INT.Ring;
    assume that
    A1: len X = len b1 and
    A2: len Y = len b1 and
    A3: for k being Nat st k in Seg len b1 holds Y.k = f.(b1/.k, v2) and
    A4: X = v1|-- b1;
    set x = v1|-- b1;
    P2: for k being Nat st k in Seg len x
    holds Y.k = (FunctionalSAF(f,v2)).(b1/.k)
    proof
      let k be Nat;
      assume k in Seg len x;
      then Y.k = f.(b1/.k,v2) by A1,A3,A4;
      hence Y.k = (FunctionalSAF(f,v2)).(b1/.k) by BLTh9;
    end;
    thus X "*" Y = (FunctionalSAF(f,v2)).(Sum lmlt(v1|--b1,b1))
    by LMThMBF1X0,A1,A2,A4,P2
    .= f.(Sum lmlt(v1|--b1,b1),v2) by BLTh9
    .= f.(v1, v2) by Th35;
  end;
