deffunc H₁( Element of X, Element of X, Element of Y, Element of Y, Element of Z, Element of Z, Element of W, Element of W) -> Element of REAL = In ((((dist ($1,$2)) + (dist ($3,$4))) + ((dist ($5,$6)) + (dist ($7,$8)))),REAL);
consider F being Function of [:[: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:],[: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:]:],REAL such that
A1: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x4, y4 being Element of W
for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = H₁(x1,y1,x2,y2,x3,y3,x4,y4) from METRIC_3:sch 3();
take F ; :: thesis: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x4, y4 being Element of W
for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
let x1, y1 be Element of X; :: thesis: for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x4, y4 being Element of W
for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
let x2, y2 be Element of Y; :: thesis: for x3, y3 being Element of Z
for x4, y4 being Element of W
for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
let x3, y3 be Element of Z; :: thesis: for x4, y4 being Element of W
for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
let x4, y4 be Element of W; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:] st x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] holds
F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
let x, y be Element of [: the carrier of X, the carrier of Y, the carrier of Z, the carrier of W:]; :: thesis: ( x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] implies F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4))) )
assume ( x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] ) ; :: thesis: F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4)))
then F . (x,y) = H₁(x1,y1,x2,y2,x3,y3,x4,y4) by A1;
hence F . (x,y) = ((dist (x1,y1)) + (dist (x2,y2))) + ((dist (x3,y3)) + (dist (x4,y4))) ; :: thesis: verum