let Al be QC-alphabet ; :: thesis: for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; :: thesis: for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ be second_Q_comp of B; :: thesis: for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B1 be Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; :: thesis: for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ1 be second_Q_comp of B1; :: thesis: ( B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) implies ( B `2 = B1 `2 & SQ = SQ1 ) )
assume that
A1: B is quantifiable and
A2: ( B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) ) ; :: thesis: ( B `2 = B1 `2 & SQ = SQ1 )
Sub_All (B,SQ) = [(All ((B `2),((B `1) `1))),SQ] by A1, SUBSTUT1:def 24;
then A3: [(All ((B `2),((B `1) `1))),SQ] = [(All ((B1 `2),((B1 `1) `1))),SQ1] by A2, SUBSTUT1:def 24;
then All ((B `2),((B `1) `1)) = All ((B1 `2),((B1 `1) `1)) by XTUPLE_0:1;
hence ( B `2 = B1 `2 & SQ = SQ1 ) by A3, QC_LANG2:5, XTUPLE_0:1; :: thesis: verum