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 Bubble-Sort-Algorithm ) +* b₁ is set & Sorting-Function . b₁ c= Result ((Initialized Bubble-Sort-Algorithm ) +* b₁) ) )
assume x in dom Sorting-Function ; :: thesis: ex b₁ being set st
( x = b₁ & (Initialized Bubble-Sort-Algorithm ) +* b₁ is set & Sorting-Function . b₁ c= Result ((Initialized Bubble-Sort-Algorithm ) +* b₁) )
then consider w being FinSequence of INT such that
A1: x = (fsloc 0 ) .--> w by Th60;
reconsider s = x as FinPartState of SCM+FSA by A1;
set p = Initialized Bubble-Sort-Algorithm ;
A2: dom s = {(fsloc 0 )} by A1, FUNCOP_1:19;
take s ; :: thesis: ( x = s & (Initialized Bubble-Sort-Algorithm ) +* s is set & Sorting-Function . s c= Result ((Initialized Bubble-Sort-Algorithm ) +* s) )
thus x = s ; :: thesis: ( (Initialized Bubble-Sort-Algorithm ) +* s is set & Sorting-Function . s c= Result ((Initialized Bubble-Sort-Algorithm ) +* s) )
A3: (Initialized Bubble-Sort-Algorithm ) +* s is autonomic by A1, Th68;
A4: dom (Initialized Bubble-Sort-Algorithm ) misses dom s by A1, Th46;

A5: now

let t be State of SCM+FSA ; :: thesis: ( (Initialized Bubble-Sort-Algorithm ) +* s c= t implies ProgramPart t halts_on t )
assume A6: (Initialized Bubble-Sort-Algorithm ) +* s c= t ; :: thesis: ProgramPart t halts_on t
set bf = bubble-sort (fsloc 0 );
Initialized Bubble-Sort-Algorithm c= (Initialized Bubble-Sort-Algorithm ) +* s by A4, FUNCT_4:33;
then Initialized Bubble-Sort-Algorithm c= t by A6, XBOOLE_1:1;
hence ProgramPart t halts_on t by Lm26, AMI_1:def 26; :: thesis: verum

end;

then A7: (Initialized Bubble-Sort-Algorithm ) +* s is halting by AMI_1:def 26;
thus (Initialized Bubble-Sort-Algorithm ) +* s is pre-program of SCM+FSA by A1, A5, Th68, AMI_1:def 26; :: thesis: Sorting-Function . s c= Result ((Initialized Bubble-Sort-Algorithm ) +* s)
consider z being FinSequence of REAL such that
A8: w,z are_fiberwise_equipotent and
A9: z is non-increasing and
z is FinSequence of INT and
A10: Sorting-Function . s = (fsloc 0 ) .--> z by A1, Th61;
consider t being State of SCM+FSA such that
A11: (Initialized Bubble-Sort-Algorithm ) +* s c= t by PBOOLE:156;
consider u being FinSequence of REAL such that
A12: w,u are_fiberwise_equipotent and
A13: u is non-increasing and
u is FinSequence of INT and
A14: (Result t) . (fsloc 0 ) = u by A1, A11, Th67;
A15: u = z by A8, A9, A12, A13, CLASSES1:84, RFINSEQ:36;
fsloc 0 in the carrier of SCM+FSA ;
then A16: fsloc 0 in dom (Result t) by PARTFUN1:def 4;
s c= (Initialized Bubble-Sort-Algorithm ) +* s by FUNCT_4:26;
then A17: dom s c= dom ((Initialized Bubble-Sort-Algorithm ) +* s) by RELAT_1:25;
A18: dom ((fsloc 0 ) .--> z) = {(fsloc 0 )} by FUNCOP_1:19;
Result ((Initialized Bubble-Sort-Algorithm ) +* s) = (Result t) | (dom ((Initialized Bubble-Sort-Algorithm ) +* s)) by A3, A7, A11, AMI_1:def 28;
hence Sorting-Function . s c= Result ((Initialized Bubble-Sort-Algorithm ) +* s) by A2, A10, A14, A15, A16, A17, A18, FUNCT_4:90, RELAT_1:186; :: thesis: verum