
theorem Th54:
  for C1, C2 being Coherence_Space for f being cap-distributive
Function of C1,C2
for x1,x2 being set st {x1,x2} in C1 for y being object st [x1,y
  ] in LinTrace f & [x2,y] in LinTrace f holds x1 = x2
proof
  let C1, C2 be Coherence_Space;
  let f be cap-distributive Function of C1,C2;
  let a1,a2 be set;
  set a = {a1,a2};
  assume
A1: a in C1;
  let y be object;
A2: a = {a1} \/ {a2} by ENUMSET1:1;
  assume [a1,y] in LinTrace f & [a2,y] in LinTrace f;
  then [{a1},y] in Trace f & [{a2},y] in Trace f by Th50;
  then {a1} = {a2} by A1,A2,Th35;
  hence thesis by ZFMISC_1:3;
end;
