
theorem
  for C1, C2 being Coherence_Space for f being U-continuous Function of
  C1,C2 for a being Element of C1 holds f.a = (graph f).:Fin a
proof
  let C1, C2 be Coherence_Space;
  let f be U-continuous Function of C1,C2;
  let a be Element of C1;
  set X = graph f;
A1: now
    let x be set;
    assume x in X;
    then
    ex y being finite set, z being set st x = [y,z] & y in dom f & z in f.
    y by Def16;
    hence x`1 is finite;
  end;
  dom f = C1 by FUNCT_2:def 1;
  then
A2: for a,b being finite Element of C1 st a c= b for y being set st [a,y] in
  X holds [b,y] in X by Th25;
  for a being finite Element of C1 for y1,y2 being set st [a,y1] in X & [a
  ,y2] in X holds {y1,y2} in C2 by Th26;
  then consider g being U-continuous Function of C1,C2 such that
A3: X = graph g and
A4: for a being Element of C1 holds g.a = X.:Fin a by A1,A2,Lm4;
  g.a = X.:Fin a by A4;
  hence thesis by A3,Th28;
end;
