let x, y, z be object ; :: thesis: for E being non empty set
for u, v being Element of E ^omega
for F being Subset of (E ^omega)
for TS being non empty transition-system over F st x,u ==>* y,TS & y,v ==>* z,TS holds
x,u ^ v ==>* z,TS
let E be non empty set ; :: thesis: for u, v being Element of E ^omega
for F being Subset of (E ^omega)
for TS being non empty transition-system over F st x,u ==>* y,TS & y,v ==>* z,TS holds
x,u ^ v ==>* z,TS
let u, v be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F st x,u ==>* y,TS & y,v ==>* z,TS holds
x,u ^ v ==>* z,TS
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F st x,u ==>* y,TS & y,v ==>* z,TS holds
x,u ^ v ==>* z,TS
let TS be non empty transition-system over F; :: thesis: ( x,u ==>* y,TS & y,v ==>* z,TS implies x,u ^ v ==>* z,TS )
assume that
A1: x,u ==>* y,TS and
A2: y,v ==>* z,TS ; :: thesis: x,u ^ v ==>* z,TS
x,(u ^ v) ^ {} ==>* y,v,TS by A1, Th96;
then A3: x,(u ^ v) ^ (<%> E) ==>* y,v ^ {},TS ;
y,v ==>* z, <%> E,TS by A2;
then x,(u ^ v) ^ {} ==>* z, <%> E,TS by A3, Th83;
hence x,u ^ v ==>* z,TS ; :: thesis: verum