let A be preIfWhileAlgebra; :: thesis: for S being non empty set
for T being Subset of S
for f being ExecutionFunction of A,S,T
for I being Element of A holds
( I is_terminating_wrt f iff I is_terminating_wrt f,S )
let S be non empty set ; :: thesis: for T being Subset of S
for f being ExecutionFunction of A,S,T
for I being Element of A holds
( I is_terminating_wrt f iff I is_terminating_wrt f,S )
let T be Subset of S; :: thesis: for f being ExecutionFunction of A,S,T
for I being Element of A holds
( I is_terminating_wrt f iff I is_terminating_wrt f,S )
let f be ExecutionFunction of A,S,T; :: thesis: for I being Element of A holds
( I is_terminating_wrt f iff I is_terminating_wrt f,S )
let I be Element of A; :: thesis: ( I is_terminating_wrt f iff I is_terminating_wrt f,S )
thus ( I is_terminating_wrt f implies I is_terminating_wrt f,S ) :: thesis: ( I is_terminating_wrt f,S implies I is_terminating_wrt f )

proof

assume A1: for s being Element of S holds [s,I] in TerminatingPrograms (A,S,T,f) ; :: according to AOFA_000:def 37 :: thesis: I is_terminating_wrt f,S
let s be Element of S; :: according to AOFA_000:def 38 :: thesis: ( s in S implies [s,I] in TerminatingPrograms (A,S,T,f) )
thus ( s in S implies [s,I] in TerminatingPrograms (A,S,T,f) ) by A1; :: thesis: verum

end;

assume A2: for s being Element of S st s in S holds
[s,I] in TerminatingPrograms (A,S,T,f) ; :: according to AOFA_000:def 38 :: thesis: I is_terminating_wrt f
let s be Element of S; :: according to AOFA_000:def 37 :: thesis: [s,I] in TerminatingPrograms (A,S,T,f)
thus [s,I] in TerminatingPrograms (A,S,T,f) by A2; :: thesis: verum