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
 dom f c= dom g & (for c st c in dom f holds f/.c = g/.c) implies f c= g
proof
  assume that
A1: dom f c= dom g and
A2: for c st c in dom f holds f/.c = g/.c;
  now
    let x be object;
    assume
A3: x in dom f;
    then reconsider x1=x as Element of C;
    f/.x1 = g/.x1 by A2,A3;
    then (f qua Function).x = g/.x1 by A3,PARTFUN1:def 6;
    hence (f qua Function).x = (g qua Function).x by A1,A3,PARTFUN1:def 6;
  end;
  hence thesis by A1,GRFUNC_1:2;
end;
