reserve i,j,k for Nat;

theorem Th8:
  for p being set holds p in dom Euclid-Function iff ex x,y being
  Integer st x > 0 & y > 0 & p = (dl.0,dl.1) --> (x,y)
proof
  let p be set;
A1: dom Euclid-Function c= FinPartSt SCM by RELAT_1:def 18;
A2: p in dom Euclid-Function iff [p,Euclid-Function.p] in Euclid-Function
  by FUNCT_1:1;
  hereby
    assume
A3: p in dom Euclid-Function;
    then Euclid-Function.p in FinPartSt SCM by PARTFUN1:4;
    then
A4: Euclid-Function.p is FinPartState of SCM by MEMSTR_0:76;
    p is FinPartState of SCM by A1,A3,MEMSTR_0:76;
    then ex x,y being Integer st x > 0 & y > 0 & p = (a,b) --> (x,y) &
    Euclid-Function.p = a .--> (x gcd y) by A2,A3,A4,Def2;
    hence ex x,y being Integer st x > 0 & y > 0 & p = (a,b) --> (x,y);
  end;
  given x,y being Integer such that
A5: x > 0 & y > 0 & p = (a,b) --> (x,y);
  [p,a .--> (x gcd y)] in Euclid-Function by A5,Def2;
  hence thesis by FUNCT_1:1;
end;
