reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th48:
  r1*(i|->r2) = i|->(r1*r2)
proof
  reconsider s1=r1, s2=r2 as Element of REAL by XREAL_0:def 1;
  s1*(i|->s2) = i|->((multreal[;](s1,id REAL)).s2) by FINSEQOP:16
    .= i|->(r1*r2) by Lm1;
  hence thesis;
end;
