 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
  for p, q being FinSequence of INT.Ring,
  p1,q1 being FinSequence of F_Real
  st p = p1 & q = q1
  holds p "*" q = p1 "*" q1
  proof
    let p, q be FinSequence of INT.Ring,
    p1, q1 be FinSequence of F_Real;
    assume AS: p = p1 & q = q1;
    A2: [:rng p,rng q:] c= [:INT,INT:] by ZFMISC_1:96;
    A3: rng <:p,q:> c= [:rng p,rng q:] by FUNCT_3:51;
    B1: dom multreal = [:REAL,REAL:] by FUNCT_2:def 1;
    [:INT,INT:] c= [:REAL,REAL:] by NUMBERS:15,ZFMISC_1:96; then
    B2: rng <:p,q:> c= dom multreal by A2,A3,B1;
    [:INT,INT:] = dom (multint) by FUNCT_2:def 1; then
    B3: rng <:p,q:> c= dom (multint) by A2,A3;
    multreal | dom (multint) = multint by LMLT12,FUNCT_2:def 1; then
    A6: multint* <:p,q:> = multreal* <:p,q:> by LMEQ5,B3,B2;
    mlt (p,q) = multint* <:p1,q1:> by AS,FUNCOP_1:def 3
    .= mlt (p1,q1) by AS,A6,FUNCOP_1:def 3;
    hence thesis by LmSign1X;
  end;
