reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th15:
  mlt(f1,(0.REAL n)+*(x,r)) = (0.REAL n)+*(x,f1.x*r)
  proof
    set p = (0.REAL n)+*(x,r);
A1: dom f1 = Seg n by FINSEQ_1:89;
A2: dom p = Seg n by FINSEQ_1:89;
A3: dom mlt(f1,p) = dom f1 /\ dom p by VALUED_1:def 4;
A4: dom ((0.REAL n)+*(x,f1.x*r)) = dom (0.REAL n) by FUNCT_7:30;
A5: dom (0.REAL n) = Seg n;
    now
      let z be object;
      assume
A6:   z in dom mlt(f1,p);
A7:   (mlt(f1,p)).z = f1.z*p.z by VALUED_1:5;
      per cases;
      suppose
A8:     z = x;
        hence (mlt(f1,p)).z = f1.z*r by A1,A2,A3,A5,A6,A7,FUNCT_7:31
        .= ((0.REAL n)+*(x,f1.x*r)).z by A1,A2,A3,A5,A6,A8,FUNCT_7:31;
      end;
      suppose
A9:     z <> x;
        hence (mlt(f1,p)).z = f1.z*(0.REAL n).z by A7,FUNCT_7:32
        .= f1.z*(n|->0).z
        .= ((0.REAL n)+*(x,f1.x*r)).z by A9,FUNCT_7:32;
      end;
    end;
    hence thesis by A4,FINSEQ_1:89;
  end;
