reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;

theorem Th48:
  E c= [:X,Y:] implies (.:pr2(X,Y)).E = pr2(X,Y).:E
proof
  assume E c= [:X,Y:];
  then E c= dom pr2(X,Y) by FUNCT_3:def 5;
  hence thesis by FUNCT_3:def 1;
end;
