let C1, C2 be Coherence_Space; :: thesis: for f being c=-monotone Function of C1,C2
for x1, x2 being object st {x1,x2} in C1 holds
for y1, y2 being object st [x1,y1] in LinTrace f & [x2,y2] in LinTrace f holds
{y1,y2} in C2
let f be c=-monotone Function of C1,C2; :: thesis: for x1, x2 being object st {x1,x2} in C1 holds
for y1, y2 being object st [x1,y1] in LinTrace f & [x2,y2] in LinTrace f holds
{y1,y2} in C2
A1: dom f = C1 by FUNCT_2:def 1;
let a1, a2 be object ; :: thesis: ( {a1,a2} in C1 implies for y1, y2 being object st [a1,y1] in LinTrace f & [a2,y2] in LinTrace f holds
{y1,y2} in C2 )
assume {a1,a2} in C1 ; :: thesis: for y1, y2 being object st [a1,y1] in LinTrace f & [a2,y2] in LinTrace f holds
{y1,y2} in C2
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 ; :: thesis: ( [a1,y1] in LinTrace f & [a2,y2] in LinTrace f implies {y1,y2} in C2 )
assume ( [a1,y1] in LinTrace f & [a2,y2] in LinTrace f ) ; :: thesis: {y1,y2} in C2
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 {y1,y2} in C2 by CLASSES1:def 1; :: thesis: verum