let H be distributive complete LATTICE; :: thesis: for a being Element of H
for X being finite Subset of H holds sup ({a} "/\" X) = a "/\" (sup X)

let a be Element of H; :: thesis: for X being finite Subset of H holds sup ({a} "/\" X) = a "/\" (sup X)
let X be finite Subset of H; :: thesis: sup ({a} "/\" X) = a "/\" (sup X)
defpred S1[ set ] means ex A being Subset of H st
( A = \$1 & a "/\" (sup A) = sup ({a} "/\" A) );
A1: S1[ {} ]
proof
reconsider A = {} as Subset of H by XBOOLE_1:2;
take A ; :: thesis: ( A = {} & a "/\" (sup A) = sup ({a} "/\" A) )
thus A = {} ; :: thesis: a "/\" (sup A) = sup ({a} "/\" A)
( Bottom H <= a & {a} "/\" ({} H) = {} ) by ;
hence a "/\" (sup A) = sup ({a} "/\" A) by YELLOW_0:25; :: thesis: verum
end;
A2: for x, B being set st x in X & B c= X & S1[B] holds
S1[B \/ {x}]
proof
let x, B be set ; :: thesis: ( x in X & B c= X & S1[B] implies S1[B \/ {x}] )
assume that
A3: x in X and
A4: B c= X and
A5: S1[B] ; :: thesis: S1[B \/ {x}]
reconsider x1 = x as Element of H by A3;
A6: {x1} c= the carrier of H ;
B c= the carrier of H by ;
then reconsider C = B \/ {x} as Subset of H by ;
take C ; :: thesis: ( C = B \/ {x} & a "/\" (sup C) = sup ({a} "/\" C) )
thus C = B \/ {x} ; :: thesis: a "/\" (sup C) = sup ({a} "/\" C)
consider A being Subset of H such that
A7: A = B and
A8: a "/\" (sup A) = sup ({a} "/\" A) by A5;
A9: {a} "/\" C = ({a} "/\" A) \/ ({a} "/\" {x1}) by
.= ({a} "/\" A) \/ {(a "/\" x1)} by YELLOW_4:46 ;
A10: ( ex_sup_of {a} "/\" A,H & ex_sup_of {(a "/\" x1)},H ) by YELLOW_0:17;
( ex_sup_of B,H & ex_sup_of {x},H ) by YELLOW_0:17;
hence a "/\" (sup C) = a "/\" (("\/" (B,H)) "\/" ("\/" ({x},H))) by YELLOW_2:3
.= (sup ({a} "/\" A)) "\/" (a "/\" ("\/" ({x},H))) by
.= (sup ({a} "/\" A)) "\/" (a "/\" x1) by YELLOW_0:39
.= (sup ({a} "/\" A)) "\/" (sup {(a "/\" x1)}) by YELLOW_0:39
.= sup ({a} "/\" C) by ;
:: thesis: verum
end;
A11: X is finite ;
S1[X] from FINSET_1:sch 2(A11, A1, A2);
hence sup ({a} "/\" X) = a "/\" (sup X) ; :: thesis: verum