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 Th74:
  Sum(F^<*r*>) = Sum F + r
proof
  reconsider s=r as Element of REAL by XREAL_0:def 1;
  reconsider F1=F as FinSequence of REAL by Lm2;
  thus Sum(F^<*r*>) = Sum(F1^<*s*>) .= addreal.(addreal $$ F1,s) by FINSOP_1:4
    .= Sum F1 + r by BINOP_2:def 9
    .= Sum F + r;
end;
