reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem Th27:
  for f,g being Function, x being set st dom f = dom g & f.x = g.x
  holds f|{x} = g|{x}
proof
  let f,g be Function, x be set such that
A1: dom f = dom g and
A2: f.x = g.x;
  per cases;
  suppose
A3: x in dom f;
    hence f|{x} = {[x,g.x]} by A2,Th26
      .= g|{x} by A1,A3,Th26;
  end;
  suppose
    not x in dom f;
    then
A4: {x} misses dom f by ZFMISC_1:50;
    hence f|{x} = {} by RELAT_1:66
      .= g|{x} by A1,A4,RELAT_1:66;
  end;
end;
