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 Th87:
  Sum(r*F) = r*(Sum F)
proof
  reconsider F1=F as FinSequence of REAL by Lm2;
  reconsider s=r as Element of REAL by XREAL_0:def 1;
  set rM = multreal[;](s,id REAL);
  thus Sum (r*F) = rM.(addreal $$ F1) by Th3,Th8,SETWOP_2:30
    .= r*(Sum F1) by Lm1
    .= r*(Sum F);
end;
