let n be non zero Nat; :: thesis: for i being Nat
for x being Point of (REAL-NS n)
for y being Element of REAL n st x = y holds
(reproj (i,y)) * (proj (1,1)) = reproj (i,x)
let i be Nat; :: thesis: for x being Point of (REAL-NS n)
for y being Element of REAL n st x = y holds
(reproj (i,y)) * (proj (1,1)) = reproj (i,x)
let x be Point of (REAL-NS n); :: thesis: for y being Element of REAL n st x = y holds
(reproj (i,y)) * (proj (1,1)) = reproj (i,x)
let y be Element of REAL n; :: thesis: ( x = y implies (reproj (i,y)) * (proj (1,1)) = reproj (i,x) )
reconsider k = proj (1,1) as Function of (REAL 1),REAL ;
A1: the carrier of (REAL-NS n) = REAL n by REAL_NS1:def 4;
assume A2: x = y ; :: thesis: (reproj (i,y)) * (proj (1,1)) = reproj (i,x)

A3: now :: thesis: for s being Element of REAL 1 holds (reproj (i,x)) . s = ((reproj (i,y)) * k) . s

let s be Element of REAL 1; :: thesis: (reproj (i,x)) . s = ((reproj (i,y)) * k) . s
reconsider r = s as Point of (REAL-NS 1) by REAL_NS1:def 4;
A4: ((reproj (i,y)) * k) . s = (reproj (i,y)) . (k . s) by FUNCT_2:15;
ex q being Element of REAL ex z being Element of REAL n st
( r = <*q*> & z = x & (reproj (i,x)) . r = (reproj (i,z)) . q ) by Def6;
hence (reproj (i,x)) . s = ((reproj (i,y)) * k) . s by A2, A4, Lm1; :: thesis: verum

end;

the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def 4;
hence (reproj (i,y)) * (proj (1,1)) = reproj (i,x) by A1, A3, FUNCT_2:63; :: thesis: verum