let x, y, z be set ; :: 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 of 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 of 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 of 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 of F st x,u ==>* y,TS & y,v ==>* z,TS holds
x,u ^ v ==>* z,TS
let TS be non empty transition-system of F; :: thesis: ( x,u ==>* y,TS & y,v ==>* z,TS implies x,u ^ v ==>* z,TS )
assume ( x,u ==>* y,TS & y,v ==>* z,TS ) ; :: thesis: x,u ^ v ==>* z,TS
then ( x,u ^ v ==>* y,v,TS & y,v ==>* z, <%> E,TS ) by ThAcc20, DefAcc;
then ( x,(u ^ v) ^ (<%> E) ==>* y,v,TS & y,v ^ (<%> E) ==>* z, <%> E,TS ) by AFINSQ_1:32;
then ( x,(u ^ v) ^ (<%> E) ==>* y,v ^ (<%> E),TS & y,v ^ (<%> E) ==>* z, <%> E,TS ) by AFINSQ_1:32;
then x,(u ^ v) ^ (<%> E) ==>* z, <%> E,TS by ThTran20;
then x,u ^ v ==>* z, <%> E,TS by AFINSQ_1:32;
hence x,u ^ v ==>* z,TS by DefAcc; :: thesis: verum