let c1, c2 be number ; :: thesis: ( ex x1, x2, y1, y2 being Element of REAL st

( x = [*x1,x2*] & y = [*y1,y2*] & c1 = [*(+ (x1,y1)),(+ (x2,y2))*] ) & ex x1, x2, y1, y2 being Element of REAL st

( x = [*x1,x2*] & y = [*y1,y2*] & c2 = [*(+ (x1,y1)),(+ (x2,y2))*] ) implies c1 = c2 )

given x1, x2, y1, y2 being Element of REAL such that A3: x = [*x1,x2*] and

A4: y = [*y1,y2*] and

A5: c1 = [*(+ (x1,y1)),(+ (x2,y2))*] ; :: thesis: ( for x1, x2, y1, y2 being Element of REAL holds

( not x = [*x1,x2*] or not y = [*y1,y2*] or not c2 = [*(+ (x1,y1)),(+ (x2,y2))*] ) or c1 = c2 )

given x19, x29, y19, y29 being Element of REAL such that A6: x = [*x19,x29*] and

A7: y = [*y19,y29*] and

A8: c2 = [*(+ (x19,y19)),(+ (x29,y29))*] ; :: thesis: c1 = c2

A9: ( x1 = x19 & x2 = x29 ) by A3, A6, ARYTM_0:10;

y1 = y19 by A4, A7, ARYTM_0:10;

hence c1 = c2 by A4, A5, A7, A8, A9, ARYTM_0:10; :: thesis: verum

( x = [*x1,x2*] & y = [*y1,y2*] & c1 = [*(+ (x1,y1)),(+ (x2,y2))*] ) & ex x1, x2, y1, y2 being Element of REAL st

( x = [*x1,x2*] & y = [*y1,y2*] & c2 = [*(+ (x1,y1)),(+ (x2,y2))*] ) implies c1 = c2 )

given x1, x2, y1, y2 being Element of REAL such that A3: x = [*x1,x2*] and

A4: y = [*y1,y2*] and

A5: c1 = [*(+ (x1,y1)),(+ (x2,y2))*] ; :: thesis: ( for x1, x2, y1, y2 being Element of REAL holds

( not x = [*x1,x2*] or not y = [*y1,y2*] or not c2 = [*(+ (x1,y1)),(+ (x2,y2))*] ) or c1 = c2 )

given x19, x29, y19, y29 being Element of REAL such that A6: x = [*x19,x29*] and

A7: y = [*y19,y29*] and

A8: c2 = [*(+ (x19,y19)),(+ (x29,y29))*] ; :: thesis: c1 = c2

A9: ( x1 = x19 & x2 = x29 ) by A3, A6, ARYTM_0:10;

y1 = y19 by A4, A7, ARYTM_0:10;

hence c1 = c2 by A4, A5, A7, A8, A9, ARYTM_0:10; :: thesis: verum