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
  f1 = f /\ g & c in dom f1 implies f1/.c = f/.c & f1/.c = g/.c
proof
  assume that
A1: f1 = f /\ g and
A2: c in dom f1;
  (f1 qua Function).c = (g qua Function).c by A1,A2,GRFUNC_1:11;
  then
A3: f1/.c = (g qua Function).c by A2,PARTFUN1:def 6;
A4: [c,(f1 qua Function).c] in f1 by A2,FUNCT_1:1;
  then [c,(f1 qua Function).c] in f by A1,XBOOLE_0:def 4;
  then
A5: c in dom f by FUNCT_1:1;
  [c,(f1 qua Function).c] in g by A1,A4,XBOOLE_0:def 4;
  then
A6: c in dom g by FUNCT_1:1;
  (f1 qua Function).c = (f qua Function).c by A1,A2,GRFUNC_1:11;
  then f1/.c = (f qua Function).c by A2,PARTFUN1:def 6;
  hence thesis by A5,A6,A3,PARTFUN1:def 6;
end;
