let x, y be object ; :: thesis: for E being non empty set
for F being Subset of (E ^omega)
for TS being non empty transition-system over 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 over 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 over 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 over 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 A1: [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 y in [: the carrier of TS,(E ^omega):] by ZFMISC_1:87;
then A2: ex y1, y2 being object st
( y1 in the carrier of TS & y2 in E ^omega & y = [y1,y2] ) by ZFMISC_1:def 2;
x in [: the carrier of TS,(E ^omega):] by A1, ZFMISC_1:87;
then ex x1, x2 being object st
( x1 in the carrier of TS & x2 in E ^omega & x = [x1,x2] ) by ZFMISC_1:def 2;
hence 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 A2; :: thesis: verum