let Al be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF Al

for X being Subset of (CQC-WFF Al) holds (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

let p, q be Element of CQC-WFF Al; :: thesis: for X being Subset of (CQC-WFF Al) holds (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

let X be Subset of (CQC-WFF Al); :: thesis: (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

reconsider pp = [((p '&' q) => (q '&' p)),5] as Element of [:(CQC-WFF Al),Proof_Step_Kinds:] by Th17, ZFMISC_1:87;

set f = <*pp*>;

A1: len <*pp*> = 1 by FINSEQ_1:40;

A2: <*pp*> . 1 = pp by FINSEQ_1:40;

then (<*pp*> . (len <*pp*>)) `1 = (p '&' q) => (q '&' p) by A1;

then A3: Effect <*pp*> = (p '&' q) => (q '&' p) by Def6;

for n being Nat st 1 <= n & n <= len <*pp*> holds

<*pp*>,n is_a_correct_step_wrt X

hence (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) } by A3; :: thesis: verum

for X being Subset of (CQC-WFF Al) holds (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

let p, q be Element of CQC-WFF Al; :: thesis: for X being Subset of (CQC-WFF Al) holds (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

let X be Subset of (CQC-WFF Al); :: thesis: (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) }

reconsider pp = [((p '&' q) => (q '&' p)),5] as Element of [:(CQC-WFF Al),Proof_Step_Kinds:] by Th17, ZFMISC_1:87;

set f = <*pp*>;

A1: len <*pp*> = 1 by FINSEQ_1:40;

A2: <*pp*> . 1 = pp by FINSEQ_1:40;

then (<*pp*> . (len <*pp*>)) `1 = (p '&' q) => (q '&' p) by A1;

then A3: Effect <*pp*> = (p '&' q) => (q '&' p) by Def6;

for n being Nat st 1 <= n & n <= len <*pp*> holds

<*pp*>,n is_a_correct_step_wrt X

proof

then
<*pp*> is_a_proof_wrt X
;
let n be Nat; :: thesis: ( 1 <= n & n <= len <*pp*> implies <*pp*>,n is_a_correct_step_wrt X )

assume ( 1 <= n & n <= len <*pp*> ) ; :: thesis: <*pp*>,n is_a_correct_step_wrt X

then A4: n = 1 by A1, XXREAL_0:1;

A5: (<*pp*> . 1) `2 = 5 by A2;

(<*pp*> . n) `1 = (p '&' q) => (q '&' p) by A2, A4;

hence <*pp*>,n is_a_correct_step_wrt X by A4, A5, Def4; :: thesis: verum

end;assume ( 1 <= n & n <= len <*pp*> ) ; :: thesis: <*pp*>,n is_a_correct_step_wrt X

then A4: n = 1 by A1, XXREAL_0:1;

A5: (<*pp*> . 1) `2 = 5 by A2;

(<*pp*> . n) `1 = (p '&' q) => (q '&' p) by A2, A4;

hence <*pp*>,n is_a_correct_step_wrt X by A4, A5, Def4; :: thesis: verum

hence (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st

( f is_a_proof_wrt X & Effect f = F ) } by A3; :: thesis: verum