set SAt = Start-At (0,SCM+FSA);
let I be Program of ; :: thesis: ( I is parahalting iff for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on s,P )

thus ( I is parahalting implies for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on s,P ) by FUNCT_4:25; :: thesis: ( ( for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on s,P ) implies I is parahalting )

assume A1: for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on s,P ; :: thesis: I is parahalting
let s be 0 -started State of SCM+FSA; :: according to AMISTD_1:def 10 :: thesis: for b1 being set holds
( not I c= b1 or b1 halts_on s )

A2: Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
let P be Instruction-Sequence of SCM+FSA; :: thesis: ( not I c= P or P halts_on s )
assume A3: I c= P ; :: thesis: P halts_on s
A4: P = P +* I by ;
A5: Initialize s = s by ;
I is_halting_on s,P by A1;
hence P halts_on s by A4, A5; :: thesis: verum