let f, g be Function; :: thesis: ( dom f = (dom f1) /\ (dom f2) & ( for x being object st x in dom f holds

f . x = (f1 . x) + (f2 . x) ) & dom g = (dom f1) /\ (dom f2) & ( for x being object st x in dom g holds

g . x = (f1 . x) + (f2 . x) ) implies f = g )

assume that

A1: dom f = (dom f1) /\ (dom f2) and

A2: for c being object st c in dom f holds

f . c = H_{1}(c)
and

A3: dom g = (dom f1) /\ (dom f2) and

A4: for c being object st c in dom g holds

g . c = H_{1}(c)
; :: thesis: f = g

f . x = (f1 . x) + (f2 . x) ) & dom g = (dom f1) /\ (dom f2) & ( for x being object st x in dom g holds

g . x = (f1 . x) + (f2 . x) ) implies f = g )

assume that

A1: dom f = (dom f1) /\ (dom f2) and

A2: for c being object st c in dom f holds

f . c = H

A3: dom g = (dom f1) /\ (dom f2) and

A4: for c being object st c in dom g holds

g . c = H

now :: thesis: for x being object st x in dom f holds

f . x = g . x

hence
f = g
by A1, A3; :: thesis: verumf . x = g . x

let x be object ; :: thesis: ( x in dom f implies f . x = g . x )

assume A5: x in dom f ; :: thesis: f . x = g . x

hence f . x = H_{1}(x)
by A2

.= g . x by A1, A3, A4, A5 ;

:: thesis: verum

end;assume A5: x in dom f ; :: thesis: f . x = g . x

hence f . x = H

.= g . x by A1, A3, A4, A5 ;

:: thesis: verum