let x0, x1, x2 be Real; :: thesis: for f being Function of REAL,REAL st x0,x1,x2 are_mutually_distinct holds
( [!f,x0,x1,x2!] = [!f,x1,x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!] )

let f be Function of REAL,REAL; :: thesis: ( x0,x1,x2 are_mutually_distinct implies ( [!f,x0,x1,x2!] = [!f,x1,x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!] ) )
set x10 = x1 - x0;
set x20 = x2 - x0;
set x12 = x1 - x2;
assume A1: x0,x1,x2 are_mutually_distinct ; :: thesis: ( [!f,x0,x1,x2!] = [!f,x1,x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!] )
then A2: x1 - x2 <> 0 by ZFMISC_1:def 5;
A3: x1 - x0 <> 0 by ;
A4: [!f,x0,x1,x2!] = ((((f . x0) - (f . x1)) / (- (x1 - x0))) - (((f . x1) - (f . x2)) / (x1 - x2))) / (- (x2 - x0))
.= ((- (((f . x0) - (f . x1)) / (x1 - x0))) - (((f . x1) - (f . x2)) / (x1 - x2))) / (- (x2 - x0)) by XCMPLX_1:188
.= (- ((((f . x0) - (f . x1)) / (x1 - x0)) + (((f . x1) - (f . x2)) / (x1 - x2)))) / (- (x2 - x0))
.= ((((f . x0) - (f . x1)) / (x1 - x0)) + (((f . x1) - (f . x2)) / (x1 - x2))) / (x2 - x0) by XCMPLX_1:191
.= (((((f . x0) - (f . x1)) * (x1 - x2)) + (((f . x1) - (f . x2)) * (x1 - x0))) / ((x1 - x0) * (x1 - x2))) / (x2 - x0) by
.= ((((f . x0) * (x1 - x2)) + ((f . x1) * (x2 - x0))) - ((f . x2) * (x1 - x0))) / (((x1 - x0) * (x1 - x2)) * (x2 - x0)) by XCMPLX_1:78 ;
A5: [!f,x2,x1,x0!] = (((- ((f . x1) - (f . x2))) / (- (x1 - x2))) - (((f . x1) - (f . x0)) / (x1 - x0))) / (x2 - x0)
.= ((((f . x1) - (f . x2)) / (x1 - x2)) - (((f . x1) - (f . x0)) / (x1 - x0))) / (x2 - x0) by XCMPLX_1:191
.= (((((f . x1) - (f . x2)) * (x1 - x0)) - (((f . x1) - (f . x0)) * (x1 - x2))) / ((x1 - x2) * (x1 - x0))) / (x2 - x0) by
.= [!f,x0,x1,x2!] by ;
x2 - x0 <> 0 by ;
then [!f,x1,x2,x0!] = (((((f . x1) - (f . x2)) * (x2 - x0)) - (((f . x2) - (f . x0)) * (x1 - x2))) / ((x1 - x2) * (x2 - x0))) / (x1 - x0) by
.= ((((f . x0) * (x1 - x2)) + ((f . x1) * (x2 - x0))) - ((f . x2) * (x1 - x0))) / (((x1 - x2) * (x2 - x0)) * (x1 - x0)) by XCMPLX_1:78
.= [!f,x0,x1,x2!] by A4 ;
hence ( [!f,x0,x1,x2!] = [!f,x1,x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!] ) by A5; :: thesis: verum