let i, k be Nat; :: thesis: for f being Element of REAL *
for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
let f be Element of REAL * ; :: thesis: for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
let r be Real; :: thesis: ( k in dom f implies ((f,i) := (k,r)) . k = r )
set fik = (f,i) := k;
A1: dom ((f,i) := k) = dom f by FUNCT_7:30;
assume k in dom f ; :: thesis: ((f,i) := (k,r)) . k = r
hence ((f,i) := (k,r)) . k = r by A1, FUNCT_7:31; :: thesis: verum