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 Th75:
  Sum(F1^F2) = Sum F1 + Sum F2
proof
  reconsider F3=F1,F4=F2 as FinSequence of REAL by Lm2;
  thus Sum(F1^F2) = Sum(F3^F4)
    .= addreal.(addreal $$ F3,addreal $$ F4) by FINSOP_1:5
    .= Sum F3 + Sum F4 by BINOP_2:def 9
    .= Sum F1 + Sum F2;
end;
