let A be non empty set ; :: thesis: for f, g, h being Element of Funcs (A,REAL)
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))

let f, g, h be Element of Funcs (A,REAL); :: thesis: for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))

let x1, x2, x3 be Element of A; :: thesis: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) implies for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h])) )

assume that
A1: A = {x1,x2,x3} and
A2: x1 <> x2 and
A3: x1 <> x3 and
A4: x2 <> x3 and
A5: f . x1 = 1 and
A6: for z being set st z in A & z <> x1 holds
f . z = 0 and
A7: g . x2 = 1 and
A8: for z being set st z in A & z <> x2 holds
g . z = 0 and
A9: h . x3 = 1 and
A10: for z being set st z in A & z <> x3 holds
h . z = 0 ; :: thesis: for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))
A11: ( g . x1 = 0 & h . x1 = 0 ) by A2, A3, A8, A10;
A12: ( f . x2 = 0 & h . x2 = 0 ) by A2, A4, A6, A10;
let h9 be Element of Funcs (A,REAL); :: thesis: ex a, b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))
take a = h9 . x1; :: thesis: ex b, c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))
take b = h9 . x2; :: thesis: ex c being Real st h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))
take c = h9 . x3; :: thesis: h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))
A13: ( f . x3 = 0 & g . x3 = 0 ) by A3, A4, A6, A8;
now :: thesis: for x being Element of A holds h9 . x = (() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x
let x be Element of A; :: thesis: h9 . x = (() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x
A14: ( x = x1 or x = x2 or x = x3 ) by ;
A15: (() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x2 = ((() . ((() . [a,f]),(() . [b,g]))) . x2) + ((() . [c,h]) . x2) by FUNCSDOM:1
.= (((() . [a,f]) . x2) + ((() . [b,g]) . x2)) + ((() . [c,h]) . x2) by FUNCSDOM:1
.= (((() . [a,f]) . x2) + ((() . [b,g]) . x2)) + (c * (h . x2)) by FUNCSDOM:4
.= (((() . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2)) by FUNCSDOM:4
.= ((a * 0) + (b * 1)) + (c * 0) by
.= h9 . x2 ;
A16: (() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x3 = ((() . ((() . [a,f]),(() . [b,g]))) . x3) + ((() . [c,h]) . x3) by FUNCSDOM:1
.= (((() . [a,f]) . x3) + ((() . [b,g]) . x3)) + ((() . [c,h]) . x3) by FUNCSDOM:1
.= (((() . [a,f]) . x3) + ((() . [b,g]) . x3)) + (c * (h . x3)) by FUNCSDOM:4
.= (((() . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3)) by FUNCSDOM:4
.= ((a * 0) + (b * 0)) + (c * 1) by
.= h9 . x3 ;
(() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x1 = ((() . ((() . [a,f]),(() . [b,g]))) . x1) + ((() . [c,h]) . x1) by FUNCSDOM:1
.= (((() . [a,f]) . x1) + ((() . [b,g]) . x1)) + ((() . [c,h]) . x1) by FUNCSDOM:1
.= (((() . [a,f]) . x1) + ((() . [b,g]) . x1)) + (c * (h . x1)) by FUNCSDOM:4
.= (((() . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1)) by FUNCSDOM:4
.= ((a * 1) + (b * 0)) + (c * 0) by
.= h9 . x1 ;
hence h9 . x = (() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x by ; :: thesis: verum
end;
hence h9 = () . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h])) by FUNCT_2:63; :: thesis: verum