let x, y be set ; :: thesis: for E being non empty set
for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [x,y] in ==>.-relation TS holds
ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] )
let E be non empty set ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [x,y] in ==>.-relation TS holds
ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] )
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F st [x,y] in ==>.-relation TS holds
ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] )
let TS be non empty transition-system of F; :: thesis: ( [x,y] in ==>.-relation TS implies ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] ) )
assume [x,y] in ==>.-relation TS ; :: thesis: ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] )
then A: ( x in [:the carrier of TS,(E ^omega ):] & y in [:the carrier of TS,(E ^omega ):] ) by ZFMISC_1:106;
consider x1, x2 being set such that
B1: ( x1 in the carrier of TS & x2 in E ^omega & x = [x1,x2] ) by A, ZFMISC_1:def 2;
consider y1, y2 being set such that
B2: ( y1 in the carrier of TS & y2 in E ^omega & y = [y1,y2] ) by A, ZFMISC_1:def 2;
thus ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( x = [s,v] & y = [t,w] ) by B1, B2; :: thesis: verum