theorem
  i in I & SF = {f} implies |:SF:|.i = {f.i}
proof
  assume that
A1: i in I and
A2: SF = {f};
A3: |:SF:| = |.SF.| by A2,Def3;
  dom |:SF:| = I by PARTFUN1:def 2;
  then i in dom f by A1,A2,A3,Th15;
  hence thesis by A2,A3,Th17;
end;
