
theorem Th25:
  for f being c=-monotone Function for a,b being set st b in dom f
& a c= b & b is finite for y being set st [a,y] in graph f holds [b,y] in graph
  f
proof
  let f be c=-monotone Function;
  let a,b be set such that
A1: b in dom f and
A2: a c= b and
A3: b is finite;
  let y be set;
  assume
A4: [a,y] in graph f;
  then a in dom f by Th24;
  then
A5: f.a c= f.b by A1,A2,Def11;
  y in f.a by A4,Th24;
  hence thesis by A1,A3,A5,Th24;
end;
