reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem Th10:
  for c being non zero Real holds lim(rseq(0,b,c,d)) = 0
  proof
    let c be non zero Real;
    set f1 = rseq(0,1,c,d);
A1: rseq(0,b,c,d) = b(#)f1
    proof
      let n be Element of NAT;
      f1.n = (0*n+1)/(c*n+d) by Th5;
      hence (b(#)f1).n = (0*n+b)/(c*n+d) by VALUED_1:6
      .= rseq(0,b,c,d).n by Th5;
    end;
    c < 0 or 0 < c;
    then lim f1 = 0 by Lm7,Lm8;
    hence 0 = b*lim f1
    .= lim(rseq(0,b,c,d)) by A1,SEQ_2:8;
  end;
