reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th26:
  f (#) <* v *> = <* f.v * v *>
proof
A1: 1 in {1} by TARSKI:def 1;
A2: len(f (#) <* v *>) = len<* v *> by Def7
    .= 1 by FINSEQ_1:40;
  then dom(f (#) <* v *>) = {1} by FINSEQ_1:2,def 3;
  then (f (#) <* v *>).1 = f.(<* v *>/.1) * <* v *>/.1 by A1,Def7
    .= f.(<* v *>/.1) * v by FINSEQ_4:16
    .= f.v * v by FINSEQ_4:16;
  hence thesis by A2,FINSEQ_1:40;
end;
