
theorem Th35:
  for C1, C2 being Coherence_Space for f being cap-distributive
Function of C1,C2 for a1,a2 being set st a1 \/ a2 in C1
 for y being object st [a1,
  y] in Trace f & [a2,y] in Trace f holds a1 = a2
proof
  let C1, C2 be Coherence_Space;
  let f be cap-distributive Function of C1,C2;
A1: dom f = C1 by FUNCT_2:def 1;
  let a1,a2 be set;
  set a = a1 \/ a2;
  assume
A2: a in C1;
  a2 c= a by XBOOLE_1:7;
  then
A3: a2 in C1 by A2,CLASSES1:def 1;
  a1 c= a by XBOOLE_1:7;
  then
A4: a1 in C1 by A2,CLASSES1:def 1;
  then reconsider b = a1 /\ a2 as Element of C1 by A3,FINSUB_1:def 2;
  b in C1;
  then
A5: C1 includes_lattice_of a1,a2 by A2,A4,A3,Th16;
  let y be object;
  assume that
A6: [a1,y] in Trace f and
A7: [a2,y] in Trace f;
  y in f.a1 & y in f.a2 by A6,A7,Th31;
  then y in (f.a1) /\ (f.a2) by XBOOLE_0:def 4;
  then
A8: y in f.b by A1,A5,Def12;
  b c= a1 by XBOOLE_1:17;
  then b c= a2 & b = a1 by A1,A6,A8,Th31,XBOOLE_1:17;
  hence thesis by A1,A7,A8,Th31;
end;
