
theorem Th53:
  for C1, C2 being Coherence_Space for f being c=-monotone
Function of C1,C2
for x1,x2 being object st {x1,x2} in C1
for y1,y2 being object st [
  x1,y1] in LinTrace f & [x2,y2] in LinTrace 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 object;
  assume {a1,a2} in C1;
  then reconsider a = {a1,a2} as Element of C1;
A2: {a2} c= a by ZFMISC_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 LinTrace f & [a2,y2] in LinTrace f;
  then
A4: y1 in f.{a1} & y2 in f.{a2} by Th52;
A5: {a1} c= a by ZFMISC_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;
