let p, q be Element of CQC-WFF ; :: thesis: for X being Subset of CQC-WFF holds (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) }
let X be Subset of CQC-WFF; :: thesis: (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,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,Proof_Step_Kinds:] by Th43, 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, MCART_1:7;
then A3: Effect <*pp*> = (p '&' q) => (q '&' p) by Def6;
for n being Element of NAT st 1 <= n & n <= len <*pp*> holds
<*pp*>,n is_a_correct_step_wrt X

proof

let n be Element of 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, MCART_1:7;
(<*pp*> . n) `1 = (p '&' q) => (q '&' p) by A2, A4, MCART_1:7;
hence <*pp*>,n is_a_correct_step_wrt X by A4, A5, Def4; :: thesis: verum

end;

then <*pp*> is_a_proof_wrt X by Def5;
hence (p '&' q) => (q '&' p) in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } by A3; :: thesis: verum