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 Th11:
  for c being non zero Real holds lim(rseq(a,b,c,d)) = a/c
  proof
    let c be non zero Real;
    set f2 = rseq(1,0,c,d);
    set f3 = rseq(0,b,c,d);
A1: rseq(a,b,c,d) = a(#)f2+f3
    proof
      let n be Element of NAT;
A2:   f2.n = (1*n+0)/(c*n+d) by Th5;
A3:   f3.n = (0*n+b)/(c*n+d) by Th5;
      (a(#)f2).n = a*f2.n by VALUED_1:6;
      hence (a(#)f2+f3).n = a*f2.n+f3.n by VALUED_1:1
      .= (a*n+b)/(c*n+d) by A2,A3
      .= rseq(a,b,c,d).n by Th5;
    end;
A4: lim f2 = 1/c by Lm13;
A5: lim f3 = 0 by Th10;
    thus lim(rseq(a,b,c,d)) = lim(a(#)f2)+lim f3 by A1,SEQ_2:6
    .= a/c by A4,A5,SEQ_2:8;
  end;
