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
  <*r1*> + <*r2*> = <*r1+r2*>
proof
  reconsider s1=r1, s2=r2 as Element of REAL by XREAL_0:def 1;
  <*s1*> + <*s2*> = <*addreal.(s1,s2)*> by FINSEQ_2:74
    .= <*r1+r2*> by BINOP_2:def 9;
  hence thesis;
end;
