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
  r*<*r1*> = <*r*r1*>
proof
  reconsider s=r, s1=r1 as Element of REAL by XREAL_0:def 1;
  s*<*s1*> = <*(multreal[;](s,id REAL)).s1*> by FINSEQ_2:35
    .= <*r*r1*> by Lm1;
  hence thesis;
end;
