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 g & f1 = f \/ g implies f1/.c = g/.c
proof
  assume that
A1: c in dom g and
A2: f1 = f \/ g;
  [c,(g qua Function).c] in g by A1,FUNCT_1:1;
  then [c,(g qua Function).c] in f1 by A2,XBOOLE_0:def 3;
  then
A3: c in dom f1 by FUNCT_1:1;
  (f1 qua Function).c = (g qua Function).c by A1,A2,GRFUNC_1:15;
  then f1/.c = (g qua Function).c by A3,PARTFUN1:def 6;
  hence thesis by A1,PARTFUN1:def 6;
end;
