
theorem
  for C1, C2 being Coherence_Space for X being Subset of [:C1, union C2
  :] st (for x being set st x in X holds x`1 is finite) & (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) & (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) ex f being U-continuous Function of C1,C2 st X = graph f
proof
  let C1, C2 be Coherence_Space;
  let X be Subset of [:C1, union C2:];
  assume
A1: not thesis;
  then ex f being U-continuous Function of C1,C2 st X = graph f & for a being
  Element of C1 holds f.a = X.:Fin a by Lm4;
  hence thesis by A1;
end;
