reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  y in R.:D1 iff ex x being Element of D st [x,y] in R & x in D1
proof
  thus y in R.:D1 implies ex x being Element of D st [x,y] in R & x in D1
  proof
    assume y in R.:D1;
    then consider x being object such that
A1: [x,y] in R and
A2: x in D1 by RELAT_1:def 13;
    reconsider a = x as Element of D by A1,ZFMISC_1:87;
    take a;
    thus thesis by A1,A2;
  end;
  given x such that
A3: [x,y] in R & x in D1;
  thus thesis by A3,RELAT_1:def 13;
end;
