let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic halting AMI-Struct of N
for p being NAT -defined the Instructions of b₁ -valued non halt-free Function
for d being FinPartState of S holds
( d is Autonomy of p iff p,d computes {} .--> {} )
let S be non empty stored-program IC-Ins-separated definite realistic halting AMI-Struct of N; :: thesis: for p being NAT -defined the Instructions of S -valued non halt-free Function
for d being FinPartState of S holds
( d is Autonomy of p iff p,d computes {} .--> {} )
let p be NAT -defined the Instructions of S -valued non halt-free Function; :: thesis: for d being FinPartState of S holds
( d is Autonomy of p iff p,d computes {} .--> {} )
let d be FinPartState of S; :: thesis: ( d is Autonomy of p iff p,d computes {} .--> {} )
thus ( d is Autonomy of p implies p,d computes {} .--> {} ) :: thesis: ( p,d computes {} .--> {} implies d is Autonomy of p )

proof

A1: dom ({} .--> {}) = {{}} by FUNCOP_1:19;
assume A2: d is Autonomy of p ; :: thesis: p,d computes {} .--> {}
let x be set ; :: according to EXTPRO_1:def 13 :: thesis: ( x in dom ({} .--> {}) implies ex s being FinPartState of S st
( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) )
assume x in dom ({} .--> {}) ; :: thesis: ex s being FinPartState of S st
( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )
then A3: x = {} by A1, TARSKI:def 1;
then reconsider s = x as FinPartState of S by FUNCT_1:174, RELAT_1:206;
take s ; :: thesis: ( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )
thus x = s ; :: thesis: ( d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )
thus d +* s is Autonomy of p by A2, A3, FUNCT_4:22; :: thesis: ({} .--> {}) . s c= Result (p,(d +* s))
({} .--> {}) . s = {} by A3, FUNCOP_1:87;
hence ({} .--> {}) . s c= Result (p,(d +* s)) by XBOOLE_1:2; :: thesis: verum

end;

dom ({} .--> {}) = {{}} by FUNCOP_1:19;
then A4: {} in dom ({} .--> {}) by TARSKI:def 1;
assume p,d computes {} .--> {} ; :: thesis: d is Autonomy of p
then ex s being FinPartState of S st
( {} = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) by A4, Def29;
hence d is Autonomy of p by FUNCT_4:22; :: thesis: verum