
theorem Th34:
  for C1, C2 being Coherence_Space for f being c=-monotone
Function of C1,C2 for a1,a2 being set st a1 \/ a2 in C1
for y1,y2 being object st
  [a1,y1] in Trace f & [a2,y2] in Trace f holds {y1,y2} in C2
proof
  let C1, C2 be Coherence_Space;
  let f be c=-monotone Function of C1,C2;
A1: dom f = C1 by FUNCT_2:def 1;
  let a1,a2 be set;
  set a = a1 \/ a2;
  assume a in C1;
  then reconsider a as Element of C1;
A2: a2 c= a by XBOOLE_1:7;
  then a2 in C1 by CLASSES1:def 1;
  then
A3: f.a2 c= f.a by A1,A2,Def11;
  let y1,y2 be object;
  assume [a1,y1] in Trace f & [a2,y2] in Trace f;
  then
A4: y1 in f.a1 & y2 in f.a2 by Th31;
A5: a1 c= a by XBOOLE_1:7;
  then a1 in C1 by CLASSES1:def 1;
  then f.a1 c= f.a by A1,A5,Def11;
  then {y1,y2} c= f.a by A3,A4,ZFMISC_1:32;
  hence thesis by CLASSES1:def 1;
end;
