let p, q be Element of CQC-WFF ; :: thesis: ( ( for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p & S `2 = Sub ) ) & ( for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = q & S `2 = Sub ) ) implies for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p '&' q & S `2 = Sub ) )
assume that
A1: for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p & S `2 = Sub ) and
A2: for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = q & S `2 = Sub ) ; :: thesis: for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p '&' q & S `2 = Sub )
let Sub be CQC_Substitution; :: thesis: ex S being Element of CQC-Sub-WFF st
( S `1 = p '&' q & S `2 = Sub )
consider S1 being Element of CQC-Sub-WFF such that
A3: ( S1 `1 = p & S1 `2 = Sub ) by A1;
consider S2 being Element of CQC-Sub-WFF such that
A4: ( S2 `1 = q & S2 `2 = Sub ) by A2;
S2 = [q,Sub] by A4, SUBSTUT1:10;
then [q,Sub] in QC-Sub-WFF ;
then A5: [(@ q),Sub] in QC-Sub-WFF by QC_LANG1:def 11;
S1 = [p,Sub] by A3, SUBSTUT1:10;
then [p,Sub] in QC-Sub-WFF ;
then [(@ p),Sub] in QC-Sub-WFF by QC_LANG1:def 11;
then [((<*[2,0]*> ^ (@ p)) ^ (@ q)),Sub] in QC-Sub-WFF by A5, SUBSTUT1:def 16;
then reconsider S = [(p '&' q),Sub] as Element of QC-Sub-WFF by QC_LANG1:def 14;
take S ; :: thesis: ( S is Element of CQC-Sub-WFF & S `1 = p '&' q & S `2 = Sub )
S `1 = p '&' q by MCART_1:7;
then S in CQC-Sub-WFF by SUBSTUT1:def 39;
then reconsider S = S as Element of CQC-Sub-WFF ;
S `2 = Sub by MCART_1:7;
hence ( S is Element of CQC-Sub-WFF & S `1 = p '&' q & S `2 = Sub ) by MCART_1:7; :: thesis: verum