
theorem
  for C1, C2 being Coherence_Space for f being U-linear Function of C1,
  C2 for a being Element of C1 holds f.a = (LinTrace f).:a
proof
  let C1, C2 be Coherence_Space;
  let f be U-linear Function of C1,C2;
  let a be Element of C1;
  set X = LinTrace f;
  ( for a,b being set 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 set st {a,b} in C1
for y being
  object st [a,y] in X & [b,y] in X holds a = b by Th53,Th54;
  then consider g being U-linear Function of C1,C2 such that
A1: X = LinTrace g and
A2: for a being Element of C1 holds g.a = X.:a by Lm6;
  g.a = X.:a by A2;
  hence thesis by A1,Th55;
end;
