let x be bound_QC-variable; :: thesis: for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
let S be Element of CQC-Sub-WFF ; :: thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
let xSQ be second_Q_comp of [S,x]; :: thesis: ( [S,x] is quantifiable implies ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) ) )
assume [S,x] is quantifiable ; :: thesis: ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
then CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = [S,x] `1 by Th30;
hence ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) ) by MCART_1:7; :: thesis: verum