
theorem Th44:
  for C1, C2 being Coherence_Space for f being U-stable Function
  of C1,C2 for X being Subset of Trace f ex g being U-stable Function of C1, C2
  st Trace g = X
proof
  let C1, C2 be Coherence_Space;
  let f be U-stable Function of C1,C2;
  let X be Subset of Trace f;
A1: for a,b be Element of C1 st a \/ b in C1
for y be object st [a,y] in X & [b
  ,y] in X holds a = b by Th35;
A2: now
    let x be set;
    assume
A3: x in X;
    then consider a, y being set such that
A4: x = [a,y] and
    a in dom f and
    y in f.a and
    for b being set st b in dom f & b c= a & y in f.b holds a = b by Def17;
    dom f = C1 by FUNCT_2:def 1;
    then a is finite by A3,A4,Th33;
    hence x`1 is finite by A4;
  end;
  X is Subset of [:C1, union C2:] & for a,b be Element of C1 st a \/ b in
  C1
for y1,y2 be object st [a,y1] in X & [b,y2] in X holds {y1,y2} in C2 by Th34,
XBOOLE_1:1;
  hence thesis by A2,A1,Th38;
end;
