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
  c in dom f & f/.c in SD iff [c,f/.c] in (SD|`f)
proof
  thus c in dom f & f/.c in SD implies [c,f/.c] in (SD|`f)
  proof
    assume that
A1: c in dom f and
A2: f/.c in SD;
    (f qua Function).c in SD by A1,A2,PARTFUN1:def 6;
    then [c,(f qua Function).c] in (SD|`f) by A1,GRFUNC_1:24;
    hence thesis by A1,PARTFUN1:def 6;
  end;
  assume [c,f/.c] in (SD|`f);
  then c in dom (SD|`f) by FUNCT_1:1;
  then c in dom f & (f qua Function).c in SD by FUNCT_1:54;
  hence thesis by PARTFUN1:def 6;
end;
