let A be non empty set ; :: thesis: for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
let x1, x2, x3, x4 be Element of A; :: thesis: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 implies ex f, g, h, f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1]) )
assume A1: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) ; :: thesis: ex f, g, h, f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
consider f being Element of Funcs A,REAL such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider g being Element of Funcs A,REAL such that
A3: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
consider h being Element of Funcs A,REAL such that
A4: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider f1 being Element of Funcs A,REAL such that
A5: ( f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) ) by Th10;
take f ; :: thesis: ex g, h, f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take g ; :: thesis: ex h, f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take h ; :: thesis: ex f1 being Element of Funcs A,REAL st
for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take f1 ; :: thesis: for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
let h' be Element of Funcs A,REAL ; :: thesis: ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
thus ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1]) by A1, A2, A3, A4, A5, Th20; :: thesis: verum