let k be Element of NAT ; :: thesis: for V being non empty Subset of QC-variables
for P being QC-pred_symbol of k
for l being QC-variable_list of k holds
( Vars ((P ! l),V) = variables_in (l,V) & Vars ((P ! l),V) = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in V ) } )
let V be non empty Subset of QC-variables; :: thesis: for P being QC-pred_symbol of k
for l being QC-variable_list of k holds
( Vars ((P ! l),V) = variables_in (l,V) & Vars ((P ! l),V) = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in V ) } )
let P be QC-pred_symbol of k; :: thesis: for l being QC-variable_list of k holds
( Vars ((P ! l),V) = variables_in (l,V) & Vars ((P ! l),V) = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in V ) } )
let l be QC-variable_list of k; :: thesis: ( Vars ((P ! l),V) = variables_in (l,V) & Vars ((P ! l),V) = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in V ) } )
A1: P ! l is atomic by QC_LANG1:def 16;
then the_arguments_of (P ! l) = l by QC_LANG1:def 21;
hence ( Vars ((P ! l),V) = variables_in (l,V) & Vars ((P ! l),V) = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in V ) } ) by A1, Th47; :: thesis: verum