let P be Instruction-Sequence of SCMPDS; :: thesis: for I, J being Program of

for s being 0 -started State of SCMPDS

for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let I, J be Program of ; :: thesis: for s being 0 -started State of SCMPDS

for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let s be 0 -started State of SCMPDS; :: thesis: for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let k be Nat; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) implies CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS )

set P1 = P +* (stop I);

set P2 = P +* (stop (I ';' J));

set m = LifeSpan ((P +* (stop I)),s);

set s3 = Comput ((P +* (stop (I ';' J))),s,k);

set P3 = P +* (stop (I ';' J));

assume that

A1: I is_closed_on s,P and

A2: I is_halting_on s,P and

A3: k < LifeSpan ((P +* (stop I)),s) ; :: thesis: CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

assume CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) = halt SCMPDS ; :: thesis: contradiction

then A4: CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = halt SCMPDS by A1, A2, A3, SCMPDS_6:27;

Initialize s = s by MEMSTR_0:44;

then P +* (stop I) halts_on s by A2, SCMPDS_6:def 3;

hence contradiction by A3, A4, EXTPRO_1:def 15; :: thesis: verum

for s being 0 -started State of SCMPDS

for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let I, J be Program of ; :: thesis: for s being 0 -started State of SCMPDS

for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let s be 0 -started State of SCMPDS; :: thesis: for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds

CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

let k be Nat; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) implies CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS )

set P1 = P +* (stop I);

set P2 = P +* (stop (I ';' J));

set m = LifeSpan ((P +* (stop I)),s);

set s3 = Comput ((P +* (stop (I ';' J))),s,k);

set P3 = P +* (stop (I ';' J));

assume that

A1: I is_closed_on s,P and

A2: I is_halting_on s,P and

A3: k < LifeSpan ((P +* (stop I)),s) ; :: thesis: CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) <> halt SCMPDS

assume CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) = halt SCMPDS ; :: thesis: contradiction

then A4: CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = halt SCMPDS by A1, A2, A3, SCMPDS_6:27;

Initialize s = s by MEMSTR_0:44;

then P +* (stop I) halts_on s by A2, SCMPDS_6:def 3;

hence contradiction by A3, A4, EXTPRO_1:def 15; :: thesis: verum