
theorem Th38:
  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
Element of C1 st a \/ b in C1
for y1,y2 being object st [a,y1] in X & [b,y2] in X
holds {y1,y2} in C2) & (for a,b being Element of C1 st a \/ b in C1
for y being object
  st [a,y] in X & [b,y] in X holds a = b) ex f being U-stable Function of C1,
  C2 st X = Trace 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-stable Function of C1,C2 st X = Trace f & for a being
  Element of C1 holds f.a = X.:Fin a by Lm5;
  hence thesis by A1;
end;
