reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem Th15:
  g = f|X iff dom g = dom f /\ X & for c st c in dom g holds g/.c = f/.c
proof
  thus g = f|X implies dom g = dom f /\ X & for c st c in dom g holds g/.c = f
  /.c
  proof
    assume
A1: g = f|X;
    hence dom g = dom f /\ X by RELAT_1:61;
    let c;
    assume
A2: c in dom g;
    then (g qua Function).c = (f qua Function).c by A1,FUNCT_1:47;
    then
A3: g/.c = (f qua Function).c by A2,PARTFUN1:def 6;
    dom g = dom f /\ X by A1,RELAT_1:61;
    then c in dom f by A2,XBOOLE_0:def 4;
    hence thesis by A3,PARTFUN1:def 6;
  end;
  assume that
A4: dom g = dom f /\ X and
A5: for c st c in dom g holds g/.c = f/.c;
  now
    let x be object;
    assume
A6: x in dom g;
    then reconsider y=x as Element of C;
    g/.y = f/.y by A5,A6;
    then
A7: (g qua Function).y = f/.y by A6,PARTFUN1:def 6;
    x in dom f by A4,A6,XBOOLE_0:def 4;
    hence (g qua Function).x = (f qua Function).x by A7,PARTFUN1:def 6;
  end;
  hence thesis by A4,FUNCT_1:46;
end;
