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
  0*R = i|->0
proof
A1: rng R c= REAL;
  thus 0*R = multreal[;](0,(id REAL)*R) by FUNCOP_1:34
    .= multreal[;](0,R) by A1,RELAT_1:53
    .= i|->0 by Th3,Th8,BINOP_2:2,FINSEQOP:76;
end;
