theorem
  c in dom f & c in SC iff [c,f/.c] in (f|SC)
proof
  thus c in dom f & c in SC implies [c,f/.c] in (f|SC)
  proof
    assume that
A1: c in dom f and
A2: c in SC;
    [c,(f qua Function).c] in (f|SC) by A1,A2,GRFUNC_1:22;
    hence thesis by A1,PARTFUN1:def 6;
  end;
  assume [c,f/.c] in (f|SC);
  then c in dom (f|SC) by FUNCT_1:1;
  then c in dom f /\ SC by RELAT_1:61;
  hence thesis by XBOOLE_0:def 4;
end;
