reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th48:
  x in X implies (f|X).x = f.x
proof
  assume
A1: x in X;
  per cases;
  suppose
    x in dom f;
    then x in dom(f|X) by A1,RELAT_1:57;
    hence thesis by Th46;
  end;
  suppose
A2: not x in dom f;
    then not x in dom(f|X) by RELAT_1:57;
    hence (f|X).x = {} by Def2
      .= f.x by A2,Def2;
  end;
end;
