let x1, x2, y1, y2, z be object ; :: thesis: for E being non empty set
for F being Subset of (E ^omega)
for TS being transition-system over F st the Tran of TS is Function & x1,x2 ==>. y1,z,TS & x1,x2 ==>. y2,z,TS holds
y1 = y2
let E be non empty set ; :: thesis: for F being Subset of (E ^omega)
for TS being transition-system over F st the Tran of TS is Function & x1,x2 ==>. y1,z,TS & x1,x2 ==>. y2,z,TS holds
y1 = y2
let F be Subset of (E ^omega); :: thesis: for TS being transition-system over F st the Tran of TS is Function & x1,x2 ==>. y1,z,TS & x1,x2 ==>. y2,z,TS holds
y1 = y2
let TS be transition-system over F; :: thesis: ( the Tran of TS is Function & x1,x2 ==>. y1,z,TS & x1,x2 ==>. y2,z,TS implies y1 = y2 )
assume A1: the Tran of TS is Function ; :: thesis: ( not x1,x2 ==>. y1,z,TS or not x1,x2 ==>. y2,z,TS or y1 = y2 )
assume that
A2: x1,x2 ==>. y1,z,TS and
A3: x1,x2 ==>. y2,z,TS ; :: thesis: y1 = y2
consider v1, w1 being Element of E ^omega such that
A4: v1 = z and
A5: x1,w1 -->. y1,TS and
A6: x2 = w1 ^ v1 by A2;
consider v2, w2 being Element of E ^omega such that
A7: v2 = z and
A8: x1,w2 -->. y2,TS and
A9: x2 = w2 ^ v2 by A3;
w1 = w2 by A4, A6, A7, A9, AFINSQ_1:28;
hence y1 = y2 by A1, A5, A8, Th17; :: thesis: verum