theorem Th20:
  x in rng p implies x..p in dom p
proof
p " {x} c= dom p & dom p = Seg(len p) by FINSEQ_1:def 3,RELAT_1:132;
    then
a1: p"{x} is included_in_Seg;
  assume x in rng p;
  then p " {x} <> {} by FUNCT_1:72;
  then rng(Sgm(p " {x})) <> {} by a1,FINSEQ_1:def 14;
  then 1 in dom(Sgm(p " {x})) by FINSEQ_3:32;
  then x..p in rng(Sgm(p " {x})) by FUNCT_1:def 3;
  then x..p in p " {x} by a1,FINSEQ_1:def 14;
  hence thesis by FUNCT_1:def 7;
end;
