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 Th5:
  rseq(a,b,c,d).n = (a*n+b)/(c*n+d)
proof
   A1: dom Re(Rat_Exp_Seq(a,b,c,d)) = NAT & n in NAT by FUNCT_2:def 1,
          ORDINAL1:def 12;
thus   rseq(a,b,c,d).n = Re (Rat_Exp_Seq(a,b,c,d).n) by COMSEQ_3:def 3,A1
.= Re (Polynom(a,b,n)/Polynom(c,d,n)) by Def1
.= Re ((a*n+b)/Polynom(c,d,n)) by POLYEQ_1:def 1
.= (a*n+b)/(c*n+d) by POLYEQ_1:def 1;
end;
