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 Th46:
  x in dom(f|X) implies (f|X).x = f.x
proof
  set g = f|X;
  assume
A1: x in dom g;
  dom g = dom f /\ X by RELAT_1:61;
  then
A2: x in dom f by A1,XBOOLE_0:def 4;
  g c= f & [x,g.x] in g by A1,Def2,RELAT_1:59;
  hence g.x = f.x by A2,Def2;
end;
