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
  (i|->r1) - (i|->r2) = i|->(r1-r2)
proof
  reconsider s1=r1, s2=r2 as Element of REAL by XREAL_0:def 1;
  (i|->s1) - (i|->s2) = i|->(diffreal.(s1,s2)) by FINSEQOP:17
    .= i|->(r1-r2) by BINOP_2:def 10;
  hence thesis;
end;
