set p = Initialized Insert-Sort-Algorithm ;
let x be set ; :: according to AMI_1:def 29 :: thesis: ( not x in proj1 Sorting-Function or ex b₁ being set st
( x = b₁ & (Initialized Insert-Sort-Algorithm ) +* b₁ is set & Sorting-Function . b₁ c= Result ((Initialized Insert-Sort-Algorithm ) +* b₁) ) )
assume x in dom Sorting-Function ; :: thesis: ex b₁ being set st
( x = b₁ & (Initialized Insert-Sort-Algorithm ) +* b₁ is set & Sorting-Function . b₁ c= Result ((Initialized Insert-Sort-Algorithm ) +* b₁) )
then consider w being FinSequence of INT such that
A1: x = (fsloc 0 ) .--> w by SCMBSORT:60;
reconsider s = x as FinPartState of SCM+FSA by A1;
consider t being State of SCM+FSA such that
A2: (Initialized Insert-Sort-Algorithm ) +* s c= t by PBOOLE:156;
A3: dom (Initialized Insert-Sort-Algorithm ) misses dom s by A1, SCMBSORT:46;

A4: now

set bf = insert-sort (fsloc 0 );
let t be State of SCM+FSA ; :: thesis: ( (Initialized Insert-Sort-Algorithm ) +* s c= t implies ProgramPart t halts_on t )
assume A5: (Initialized Insert-Sort-Algorithm ) +* s c= t ; :: thesis: ProgramPart t halts_on t
Initialized Insert-Sort-Algorithm c= (Initialized Insert-Sort-Algorithm ) +* s by A3, FUNCT_4:33;
then Initialized Insert-Sort-Algorithm c= t by A5, XBOOLE_1:1;
hence ProgramPart t halts_on t by Lm12, AMI_1:def 26; :: thesis: verum

end;

then A6: (Initialized Insert-Sort-Algorithm ) +* s is halting by AMI_1:def 26;
take s ; :: thesis: ( x = s & (Initialized Insert-Sort-Algorithm ) +* s is set & Sorting-Function . s c= Result ((Initialized Insert-Sort-Algorithm ) +* s) )
thus x = s ; :: thesis: ( (Initialized Insert-Sort-Algorithm ) +* s is set & Sorting-Function . s c= Result ((Initialized Insert-Sort-Algorithm ) +* s) )
s c= (Initialized Insert-Sort-Algorithm ) +* s by FUNCT_4:26;
then A7: dom s c= dom ((Initialized Insert-Sort-Algorithm ) +* s) by RELAT_1:25;
thus (Initialized Insert-Sort-Algorithm ) +* s is pre-program of SCM+FSA by A1, A4, Th48, AMI_1:def 26; :: thesis: Sorting-Function . s c= Result ((Initialized Insert-Sort-Algorithm ) +* s)
fsloc 0 in the carrier of SCM+FSA ;
then A8: fsloc 0 in dom (Result t) by PARTFUN1:def 4;
(Initialized Insert-Sort-Algorithm ) +* s is autonomic by A1, Th48;
then A9: Result ((Initialized Insert-Sort-Algorithm ) +* s) = (Result t) | (dom ((Initialized Insert-Sort-Algorithm ) +* s)) by A6, A2, AMI_1:def 28;
A10: dom s = {(fsloc 0 )} by A1, FUNCOP_1:19;
consider u being FinSequence of REAL such that
A11: w,u are_fiberwise_equipotent and
A12: u is non-increasing and
u is FinSequence of INT and
A13: (Result t) . (fsloc 0 ) = u by A1, A2, Th47;
consider z being FinSequence of REAL such that
A14: w,z are_fiberwise_equipotent and
A15: z is non-increasing and
z is FinSequence of INT and
A16: Sorting-Function . s = (fsloc 0 ) .--> z by A1, SCMBSORT:61;
A17: dom ((fsloc 0 ) .--> z) = {(fsloc 0 )} by FUNCOP_1:19;
u = z by A14, A15, A11, A12, CLASSES1:84, RFINSEQ:36;
hence Sorting-Function . s c= Result ((Initialized Insert-Sort-Algorithm ) +* s) by A10, A16, A13, A8, A7, A17, A9, FUNCT_4:90, RELAT_1:186; :: thesis: verum