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,e] in (s*f) implies [c,f/.c] in f & [f/.c,e] in s
proof
  assume
A1: [c,e] in (s*f);
  then
A2: [(f qua Function).c,e] in s by GRFUNC_1:4;
A3: [c,(f qua Function).c] in f by A1,GRFUNC_1:4;
  then c in dom f by FUNCT_1:1;
  hence thesis by A3,A2,PARTFUN1:def 6;
end;
