reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;

theorem Th19:
  x in rng p implies p.(x..p) = x
proof
  set q = Sgm(p " {x});
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 q <> {} by a1,FINSEQ_1:def 14;
  then 1 in dom q by FINSEQ_3:32;
  then
A2: q.1 in rng q by FUNCT_1:def 3;
  rng q = p " {x} by a1,FINSEQ_1:def 14;
  then p.(x..p) in {x} by A2,FUNCT_1:def 7;
  hence thesis by TARSKI:def 1;
end;
